#6519800

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,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, 6426996);