#7185524

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#30463	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1839 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)(6,10)(8,11)(13,14)(15,20)(16,26)(17,22)(18,32)(19,23)(21,28)(24,36)(25,33)(27,40)(29,35)(30,37)(31,38)(34,41)(39,43)(42,44);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(13,15,19,18)(14,20,23,32)(16,24,29,28)(17,25,31,30)(21,26,36,35)(22,33,38,37)(27,39,42,41)(34,40,43,44);(5,7)(6,11)(8,10)(9,12)(13,16,18,28,19,29,15,24)(14,21,20,26,23,36,32,35)(17,27,30,41,31,42,25,39)(22,34,33,40,38,43,37,44);(2,4)(6,10)(7,12)(13,17,19,31)(14,22,23,38)(15,25,18,30)(16,27,29,42)(20,33,32,37)(21,34,36,43)(24,39,28,41)(26,40,35,44);(5,8,9,11)(6,7,10,12)(13,15,19,18)(14,20,23,32)(16,28,29,24)(17,30,31,25)(21,35,36,26)(22,37,38,33)(27,39,42,41)(34,40,43,44)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=i,e^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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092720);