#6464032

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371228);