#6466199

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2170 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)(13,14,20,17,19,24,18,23)(15,25,34,21,28,40,27,37)(16,29,35,39,32,22,33,31)(26,41,43,44,42,38,30,36);(2,4)(6,9)(10,11)(13,15,19,28)(14,21,24,37)(16,30,32,43)(17,25,23,40)(18,34,20,27)(22,38,29,41)(26,35,42,33)(31,44,39,36);(5,7,8,12)(6,10,9,11)(13,16,20,35,19,32,18,33)(14,22,17,31,24,29,23,39)(15,26,34,43,28,42,27,30)(21,36,40,41,37,44,25,38);(13,17,20,24,19,23,18,14)(15,21,34,40,28,37,27,25)(16,31,35,29,32,39,33,22)(26,36,43,41,42,44,30,38);(16,32)(22,29)(26,42)(30,43)(31,39)(33,35)(36,44)(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,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

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, 6373395);