#7503070

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1834 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)(13,14,18,22)(15,24,27,20)(16,29,17,31)(19,34,33,23)(21,32,37,30)(25,40,26,42)(28,36,44,39)(35,41,38,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,25)(17,26)(18,27)(19,28)(21,35)(22,24)(23,36)(29,42)(30,41)(31,40)(32,43)(33,44)(34,39)(37,38);(2,4)(13,16,18,17)(14,21,22,37)(15,25,27,26)(19,32,33,30)(20,35,24,38)(23,31,34,29)(28,43,44,41)(36,40,39,42);(6,11)(7,12)(13,17,18,16)(14,21,22,37)(15,26,27,25)(19,30,33,32)(20,35,24,38)(23,31,34,29)(28,41,44,43)(36,40,39,42);(5,8,9,10)(6,7,11,12)(15,27)(16,17)(20,24)(21,37)(28,44)(29,31)(30,32)(36,39)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,e^c=ei,g^c=gi,e^d=ei,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, 7410266);