#7054581

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	#23863	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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=df,e^b=eh,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, 6961777);