#7021865

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

2205 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)(13,14,19,34,18,24,17,23)(15,21,33,20,26,39,27,37)(16,28,35,41,32,22,31,42)(25,40,43,44,30,38,29,36);(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,20,24,37)(16,29,32,43)(17,33,19,27)(21,23,39,34)(22,38,28,40)(25,35,30,31)(36,42,44,41);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,30)(17,27)(18,26)(19,33)(20,34)(22,40)(23,37)(24,39)(25,32)(28,38)(29,35)(31,43)(36,41)(42,44);(7,12)(9,11)(13,16,19,35,18,32,17,31)(14,22,34,42,24,28,23,41)(15,25,33,43,26,30,27,29)(20,36,39,40,37,44,21,38);(6,10)(9,11)(15,26)(20,37)(21,39)(25,30)(27,33)(29,43)(36,44)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6929061);