#7021625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2213 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=fi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6928821);