#382949

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14350
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

599 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,15,20,42)(4,21)(5,11)(6,12)(7,13)(8,14)(9,18,33,19)(10,24)(16,44,46,31)(17,43,50,62)(22,34,51,57)(23,52)(25,37)(26,36)(27,35)(28,39)(29,40)(30,41)(32,48,59,49)(38,54)(45,63,55,56)(47,60,64,58)(53,61);(1,3,7,19,8,20,6,18)(2,9,12,15,14,33,13,42)(4,22,24,46,26,51,25,16)(5,17,29,49,30,50,28,48)(10,34,21,44,37,57,36,31)(11,32,39,43,41,59,40,62)(23,47,27,55,54,64,53,45)(35,60,52,63,61,58,38,56);(1,4)(2,10,14,37)(3,16,20,46)(5,27)(6,25)(7,24)(8,26)(9,31)(11,38,41,52)(12,21,13,36)(15,34)(17,47,50,64)(18,51,19,22)(23,28)(29,54)(30,53)(32,58)(33,44)(35,40,61,39)(42,57)(43,56)(45,49,55,48)(59,60)(62,63);(1,5,6,28,8,30,7,29)(2,11,12,39,14,41,13,40)(3,17,18,48,20,50,19,49)(4,23,24,27,26,54,25,53)(9,32,15,43,33,59,42,62)(10,35,36,38,37,61,21,52)(16,45,22,47,46,55,51,64)(31,56,57,58,44,63,34,60)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290145);