#7054901

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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2095 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)(13,14,19,17)(15,23,26,25)(16,21,28,31)(18,32,34,22)(20,33,30,27)(24,37,39,40)(29,42,41,36)(35,43,44,38);(5,6)(7,11)(8,10)(9,12)(13,15,19,26)(14,20,17,30)(16,24,28,39)(18,33,34,27)(21,35,31,44)(22,25,32,23)(29,43,41,38)(36,40,42,37);(2,4)(5,7,8,12)(6,9,10,11)(13,16)(14,21)(15,24)(17,31)(18,29)(19,28)(20,35)(22,36)(23,37)(25,40)(26,39)(27,38)(30,44)(32,42)(33,43)(34,41);(7,12)(9,11)(13,17,19,14)(15,25,26,23)(16,21,28,31)(18,22,34,32)(20,27,30,33)(24,37,39,40)(29,42,41,36)(35,43,44,38);(6,10)(9,11)(15,26)(16,28)(20,30)(21,31)(23,25)(27,33)(29,41)(36,42)

Polycyclic presentation:

2^9|a^2=i,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=dfi,e^b=ehi,f^b=fi,d^c=dhi,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, 6962097);