#6456574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1904 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:

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

Polycyclic presentation:

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

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, 6363770);