#7142928

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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1984 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)(7,11)(10,12)(13,14)(15,24)(16,28)(17,22)(18,32)(19,23)(20,26)(21,30)(25,40)(27,35)(29,38)(31,37)(33,36)(34,41)(39,43)(42,44);(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,23,36)(16,25,30,43)(17,26,18,33)(21,34,28,44)(22,35,32,24)(29,41,31,42)(37,39,38,40);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,31)(18,29)(19,30)(20,34)(22,38)(23,28)(24,39)(26,42)(27,43)(32,37)(33,41)(35,40)(36,44);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,23,32)(15,26,27,33)(16,29,30,31)(20,35,36,24)(21,37,28,38)(25,41,43,42)(34,39,44,40)

Polycyclic presentation:

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

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, 7050124);