#6520009

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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1764 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)(13,14,18,23,19,21,17,31)(15,24,26,34,27,36,33,20)(16,28,29,37,30,22,32,38)(25,40,42,35,43,39,41,44);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,21,34)(16,25,30,43)(17,26,18,33)(22,35,28,44)(23,36,31,24)(29,41,32,42)(37,39,38,40);(1,2,3,4)(6,9)(7,11)(14,21)(20,34)(22,28)(23,31)(24,36)(35,44)(37,38)(39,40);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,25)(17,32)(18,29)(19,30)(20,35)(21,28)(23,38)(24,39)(26,42)(27,43)(31,37)(33,41)(34,44)(36,40);(6,9)(10,12)(13,17,19,18)(14,23,21,31)(15,26,27,33)(16,29,30,32)(20,36,34,24)(22,37,28,38)(25,41,43,42)(35,39,44,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=h,e^2=i,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, 6427205);