#6466070

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2016 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:

(5,6,8,10)(7,11,12,9)(13,14,19,23)(15,24,28,39)(16,29,17,21)(18,32,34,22)(20,33,36,27)(25,41,26,40)(30,38,31,44)(35,42,37,43);(2,3)(13,15,19,28)(14,20,23,36)(16,26,17,25)(18,33,34,27)(21,37,29,35)(22,39,32,24)(30,42,31,43)(38,40,44,41);(1,2,4,3)(5,7,8,12)(6,9,10,11)(13,16,19,17)(14,21,23,29)(15,25,28,26)(18,30,34,31)(20,35,36,37)(22,38,32,44)(24,40,39,41)(27,42,33,43);(1,3,4,2)(13,17,19,16)(14,21,23,29)(15,26,28,25)(18,31,34,30)(20,35,36,37)(22,38,32,44)(24,40,39,41)(27,43,33,42);(16,17)(21,29)(25,26)(30,31)(35,37)(38,44)(40,41)(42,43)

Polycyclic presentation:

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

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, 6373266);