#6996378

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2334 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:

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

Polycyclic presentation:

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

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, 6903574);