#6520565

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2234 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)(13,14,20,26)(15,22,29,36)(16,21,28,37)(17,23,32,40)(18,25,19,24)(27,38,43,39)(30,35,31,44)(33,42,34,41);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,28,20,29,18,16)(14,21,25,36,26,37,24,22)(17,27,34,31,32,43,33,30)(23,35,42,39,40,44,41,38);(6,9)(7,11)(13,16,18,29,20,28,19,15)(14,22,24,37,26,36,25,21)(17,31,33,27,32,30,34,43)(23,39,41,35,40,38,42,44);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,27)(16,30)(18,33)(19,34)(20,32)(21,35)(22,38)(24,41)(25,42)(26,40)(28,31)(29,43)(36,39)(37,44);(6,9)(10,12)(17,32)(23,40)(27,43)(30,31)(33,34)(35,44)(38,39)(41,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6427761);