#6373540

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,20)(17,22)(18,33)(19,24)(21,28)(23,26)(27,39)(29,34)(30,35)(31,37)(32,43)(36,41)(38,40)(42,44);(6,9)(10,12)(13,15,26,30,19,28,18,16)(14,20,33,21,24,35,23,25)(17,27,40,42,31,41,32,29)(22,34,43,36,37,44,38,39);(5,7,8,11)(6,10,9,12)(13,16,18,28,19,30,26,15)(14,21,23,20,24,25,33,35)(17,29,32,41,31,42,40,27)(22,36,38,34,37,39,43,44);(1,3,4,2)(13,17)(14,22)(15,27)(16,29)(18,32)(19,31)(20,34)(21,36)(23,38)(24,37)(25,39)(26,40)(28,41)(30,42)(33,43)(35,44);(17,31)(22,37)(27,41)(29,42)(32,40)(34,44)(36,39)(38,43)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280736);