#6520005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(13,14,17,22,18,21,31,36)(15,23,25,33,26,37,30,19)(16,27,29,43,28,20,34,35)(24,39,41,32,40,38,44,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,21,33)(16,24,28,40)(17,30,31,25)(20,32,27,42)(22,37,36,23)(29,44,34,41)(35,39,43,38);(1,2,3,4)(6,9)(7,11);(5,7,8,11)(6,10,9,12)(13,16,18,28)(14,20,21,27)(15,24,26,40)(17,29,31,34)(19,32,33,42)(22,35,36,43)(23,38,37,39)(25,41,30,44);(6,9)(10,12)(14,21)(16,28)(19,33)(22,36)(23,37)(24,40)(29,34)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6427201);