#7054109

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

2139 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,20)(17,30)(18,21)(19,26)(22,32)(24,38)(25,39)(27,41)(28,37)(29,44)(31,33)(34,40)(35,42)(36,43);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(15,24)(19,33)(23,38)(25,27)(26,31)(34,35)(39,41)(40,42);(5,7,8,12)(6,9,10,11)(13,15,18,24)(14,19,21,33)(16,27,28,25)(17,31,32,26)(20,35,37,34)(22,38,30,23)(29,40,43,42)(36,39,44,41);(2,4)(7,12)(9,11)(13,16,18,28)(14,20,21,37)(15,25,24,27)(17,29,32,43)(19,34,33,35)(22,36,30,44)(23,39,38,41)(26,40,31,42);(6,10)(9,11)(14,21)(15,24)(16,28)(22,30)(26,31)(29,43)(34,35)(39,41)

Polycyclic presentation:

2^9|c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961305);