#6466007

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1982 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)(13,14,18,23)(15,20,27,37)(16,29,31,21)(17,22,32,41)(19,24,30,40)(25,43,33,35)(26,36,44,39)(28,38,34,42);(2,4)(13,15,18,27)(14,20,23,37)(16,26,31,44)(17,33,32,25)(19,34,30,28)(21,39,29,36)(22,35,41,43)(24,42,40,38);(5,7,8,12)(6,9,10,11)(13,16,30,17,18,31,19,32)(14,21,40,41,23,29,24,22)(15,25,34,44,27,33,28,26)(20,35,42,36,37,43,38,39);(13,17,30,31,18,32,19,16)(14,22,40,21,23,41,24,29)(15,26,34,25,27,44,28,33)(20,36,42,43,37,39,38,35);(14,23)(20,37)(21,29)(22,41)(24,40)(35,43)(36,39)(38,42)

Polycyclic presentation:

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

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, 6373203);