#7503695

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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 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)(7,9)(8,10)(11,12)(13,14,17,22,18,23,30,20)(15,19,26,35,25,34,41,32)(16,27,28,36,29,37,31,21)(24,38,40,43,39,44,42,33);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15)(14,19)(16,24)(17,26)(18,25)(20,32)(21,33)(22,35)(23,34)(27,38)(28,40)(29,39)(30,41)(31,42)(36,43)(37,44);(2,4)(5,8,9,12)(6,10,7,11)(14,20)(17,30)(19,32)(21,27)(22,23)(26,41)(28,31)(33,38)(34,35)(36,37)(40,42)(43,44);(6,7)(10,11)(13,16,18,29)(14,21,23,36)(15,24,25,39)(17,31,30,28)(19,33,34,43)(20,27,22,37)(26,42,41,40)(32,38,35,44);(15,25)(19,34)(24,39)(26,41)(32,35)(33,43)(38,44)(40,42)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7410891);