#6574428

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

1945 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)(13,14,17,21)(15,23,26,19)(16,20,30,39)(18,22,32,41)(24,43,44,34)(25,36,27,35)(28,38,29,37)(31,40,33,42);(5,7,8,12)(6,9,10,11)(13,15,17,26)(14,19,21,23)(16,28,30,29)(18,25,32,27)(20,37,39,38)(22,35,41,36)(24,33,44,31)(34,42,43,40);(2,4)(13,16,17,30)(14,20,21,39)(15,24,26,44)(18,33,32,31)(19,34,23,43)(22,42,41,40)(25,28,27,29)(35,37,36,38);(13,15,17,26)(14,19,21,23)(16,29,30,28)(18,25,32,27)(20,38,39,37)(22,35,41,36)(24,31,44,33)(34,40,43,42);(15,25)(18,32)(19,35)(22,41)(23,36)(24,28)(26,27)(29,44)(31,33)(34,37)(38,43)(40,42)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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, 6481624);