#7154761

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2071 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)(13,14,19,23)(15,20,18,33)(16,26,28,42)(17,22,31,38)(21,24,36,29)(25,34,32,44)(27,41,30,43)(35,39,37,40);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,23,33)(16,24,28,29)(17,25,31,32)(21,26,36,42)(22,34,38,44)(27,39,30,40)(35,41,37,43);(5,6)(7,10)(8,9)(11,12)(13,16,18,29,19,28,15,24)(14,21,20,26,23,36,33,42)(17,30,32,39,31,27,25,40)(22,37,34,43,38,35,44,41);(2,4)(6,9)(10,12)(13,17,19,31)(14,22,23,38)(15,25,18,32)(16,27,28,30)(20,34,33,44)(21,35,36,37)(24,39,29,40)(26,41,42,43);(5,7,8,11)(6,10,9,12)(16,28)(17,31)(21,36)(22,38)(24,29)(25,32)(26,42)(34,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=h,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,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, 7061957);