#6477128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,7)(3,8,4,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,31,39)(20,32,34,25)(21,26,22,36)(24,37,40,30)(29,43,45,42)(33,47,35,41)(38,44,46,48);(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,27,40)(20,29,34,45)(21,30,22,37)(25,38,32,46)(26,39,36,28)(33,44,35,48)(41,43,47,42);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,38)(26,41)(27,32)(28,42)(30,44)(31,45)(36,47)(37,48)(39,43)(40,46);(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,33,34,35)(24,39,40,28)(25,41,32,47)(29,44,45,48)(38,43,46,42);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(20,34)(25,32)(29,45)(33,35)(38,46)(41,47)(42,43)(44,48)

Polycyclic presentation:

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

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, 6384324);