#7062697

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
prime radical	#42797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1855 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)(14,19)(15,23)(17,29)(20,34)(21,33)(22,30)(24,38)(26,37)(27,31)(32,36)(35,40)(43,44);(5,6)(7,11)(8,10)(9,12)(13,14,18,22)(15,20,23,36)(16,26,28,33)(17,30,29,19)(21,31,37,27)(24,40,38,43)(25,34,39,32)(35,42,44,41);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,24)(17,25)(18,23)(19,32)(21,35)(22,36)(26,40)(27,41)(28,38)(29,39)(30,34)(31,42)(33,43)(37,44);(7,12)(9,11)(13,16,18,28)(14,21,22,37)(15,24,23,38)(17,31,29,27)(19,33,30,26)(20,35,36,44)(25,42,39,41)(32,43,34,40);(6,10)(9,11)(14,22)(15,23)(19,30)(21,37)(24,38)(25,39)(26,33)(41,42)

Polycyclic presentation:

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

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, 6969893);