#7502725

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1818 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409921);