#470882

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16706
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

688 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378078);