#6996651

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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2330 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)(13,14,18,23,19,22,33,40)(15,24,27,36,26,41,17,20)(16,28,32,44,31,21,37,39)(25,43,30,35,29,42,34,38);(5,6)(7,11)(8,10)(9,12)(13,15,19,26)(14,20,22,36)(16,29,31,25)(17,33,27,18)(21,38,28,35)(23,41,40,24)(30,37,34,32)(39,42,44,43);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,31)(14,21,22,28)(15,25,26,29)(17,34,27,30)(18,32,33,37)(20,35,36,38)(23,39,40,44)(24,42,41,43);(7,12)(9,11)(13,17,18,15,19,27,33,26)(14,20,23,24,22,36,40,41)(16,30,32,29,31,34,37,25)(21,38,39,43,28,35,44,42);(6,10)(9,11)(14,22)(15,26)(16,31)(17,27)(23,40)(32,37)(35,38)(42,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,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=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, 6903847);