#7053999

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
prime radical	#45245	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2120 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)(13,14,18,22)(15,23,25,36)(16,20,27,34)(17,29,31,21)(19,30,33,26)(24,37,39,44)(28,41,40,35)(32,42,43,38);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7,8,12)(6,9,10,11)(13,15,18,25)(14,19,22,33)(16,24,27,39)(17,30,31,26)(20,32,34,43)(21,36,29,23)(28,42,40,38)(35,44,41,37);(2,4)(7,12)(9,11)(13,16)(14,20)(15,24)(17,28)(18,27)(19,32)(21,35)(22,34)(23,37)(25,39)(26,38)(29,41)(30,42)(31,40)(33,43)(36,44);(6,10)(9,11)(15,25)(16,27)(19,33)(20,34)(23,36)(26,30)(28,40)(35,41)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,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=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, 6961195);