#6444540

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2320 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351736);