#6996531

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

2306 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903727);