#7056297

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
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1979 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)(13,14,17,23,18,22,21,38)(15,24,28,35,27,26,33,19)(16,20,32,37,31,30,39,44)(25,40,41,36,29,42,43,34);(5,6)(7,11)(8,10)(9,12)(13,15,18,27)(14,19,22,35)(16,29,31,25)(17,33,21,28)(20,36,30,34)(23,26,38,24)(32,41,39,43)(37,40,44,42);(2,4)(5,7,8,12)(6,9,10,11)(13,16,18,31)(14,20,22,30)(15,25,27,29)(17,32,21,39)(19,34,35,36)(23,37,38,44)(24,40,26,42)(28,41,33,43);(7,12)(9,11)(13,14,21,38,18,22,17,23)(15,26,33,35,27,24,28,19)(16,30,39,37,31,20,32,44)(25,40,43,34,29,42,41,36);(6,10)(9,11)(14,22)(15,27)(16,31)(23,38)(28,33)(32,39)(34,36)(40,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6963493);