#6574418

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1943 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)(5,6,8,10)(7,11,12,9)(13,14,18,22)(15,19,26,36)(16,20,30,40)(17,21,32,42)(23,33,43,44)(24,34,25,35)(27,37,28,38)(29,39,31,41);(5,7,8,12)(6,9,10,11)(13,15,18,26)(14,19,22,36)(16,27,30,28)(17,24,32,25)(20,37,40,38)(21,34,42,35)(23,31,43,29)(33,41,44,39);(2,4)(13,16,18,30)(14,20,22,40)(15,23,26,43)(17,31,32,29)(19,33,36,44)(21,41,42,39)(24,27,25,28)(34,37,35,38);(13,15,18,26)(14,19,22,36)(16,28,30,27)(17,24,32,25)(20,38,40,37)(21,34,42,35)(23,29,43,31)(33,39,44,41);(15,24)(17,32)(19,34)(21,42)(23,27)(25,26)(28,43)(29,31)(33,37)(35,36)(38,44)(39,41)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6481614);