#6458529

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365725);