#6466188

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2186 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)(7,11)(8,9)(10,12)(13,14,20,26,19,24,18,25)(15,27,32,21,31,40,30,39)(16,33,35,42,34,22,17,23)(28,41,29,43,36,38,44,37);(2,4)(6,9)(10,11)(13,15,19,31)(14,21,24,39)(16,29,34,44)(17,36,35,28)(18,32,20,30)(22,41,33,38)(23,37,42,43)(25,40,26,27);(5,7,8,12)(6,10,9,11)(13,16,18,17,19,34,20,35)(14,22,25,42,24,33,26,23)(15,28,30,44,31,36,32,29)(21,37,27,38,39,43,40,41);(13,17,18,34,19,35,20,16)(14,23,25,22,24,42,26,33)(15,29,30,28,31,44,32,36)(21,38,27,43,39,41,40,37);(14,24)(21,39)(22,33)(23,42)(25,26)(27,40)(37,43)(38,41)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gh,d^2=hi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,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, 6373384);