#299242

order	512
nilpotency class	3
isoclinism	isoclinic class of #299233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206438);