#7156176

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2060 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)(7,11)(10,12)(13,14,20,32,19,17,18,23)(15,24,33,21,27,37,26,30)(16,22,34,44,28,31,29,36)(25,38,42,35,40,39,41,43);(6,9)(7,11)(13,15,19,27)(14,21,17,30)(16,25,28,40)(18,33,20,26)(22,35,31,43)(23,37,32,24)(29,42,34,41)(36,39,44,38);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,28)(14,22,17,31)(15,25,27,40)(18,29,20,34)(21,35,30,43)(23,36,32,44)(24,38,37,39)(26,41,33,42);(6,9)(10,12)(13,17,20,23,19,14,18,32)(15,24,33,21,27,37,26,30)(16,22,34,44,28,31,29,36)(25,39,42,43,40,38,41,35);(5,7,8,11)(6,10,9,12)(16,28)(22,31)(25,40)(29,34)(35,43)(36,44)(38,39)(41,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gi,d^2=fi,e^2=h,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, 7063372);