#7502789

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1819 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)(15,20,16,27)(17,30,32,40)(18,33,35,23)(21,26,38,29)(22,34,39,31)(25,41,28,37)(36,43,44,42);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,16)(14,20,24,27)(17,25,32,28)(18,26,35,29)(21,33,38,23)(22,36,39,44)(30,41,40,37)(31,42,34,43);(2,4)(13,16,19,15)(14,21,24,38)(17,28,32,25)(18,26,35,29)(20,33,27,23)(22,37,39,41)(30,44,40,36)(31,42,34,43);(6,11)(7,12)(13,17,19,32)(14,22,24,39)(15,25,16,28)(18,34,35,31)(20,36,27,44)(21,37,38,41)(23,40,33,30)(26,43,29,42);(5,8,9,10)(6,7,11,12)(15,16)(20,27)(21,38)(25,28)(26,29)(36,44)(37,41)(42,43)

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,e^c=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, 7409985);