#6454455

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1952 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)(7,10)(8,9)(11,12);(2,4)(13,14,18,22)(15,23,25,37)(16,27,29,38)(17,30,33,21)(19,31,35,26)(20,32,36,28)(24,34,40,43)(39,42,44,41);(6,9)(10,12)(13,15,18,25)(14,19,22,35)(16,24,29,40)(17,31,33,26)(20,34,36,43)(21,37,30,23)(27,39,38,44)(28,42,32,41);(13,16,18,29)(14,20,22,36)(15,24,25,40)(17,32,33,28)(19,34,35,43)(21,38,30,27)(23,39,37,44)(26,42,31,41);(5,7,8,11)(6,10,9,12)(15,25)(19,35)(23,37)(24,40)(26,31)(34,43)(39,44)(41,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6361651);