#6467197

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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

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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6374393);