#6463000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2364 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)(5,6)(7,12)(8,10)(9,11)(13,14,17,21,18,22,31,38)(15,23,26,33,25,37,30,19)(16,20,29,36,28,35,43,44)(24,39,40,34,27,41,42,32);(7,8)(10,12)(13,15)(14,19)(16,27)(17,30)(18,25)(20,34)(21,37)(22,33)(23,38)(24,28)(26,31)(29,40)(32,35)(36,39)(41,44)(42,43);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,32)(21,36)(22,35)(23,39)(25,27)(26,40)(30,42)(31,43)(33,34)(37,41)(38,44);(5,8,9,7)(6,10,11,12)(15,25)(19,33)(23,37)(24,27)(26,30)(32,34)(39,41)(40,42);(16,28)(20,35)(24,27)(29,43)(32,34)(36,44)(39,41)(40,42)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,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, 6370196);