#6516893

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3284 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=hi,e^2=i,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^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6424089);