#6435500

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432843
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2004 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342696);