#6475494

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1727 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,28,32,43)(20,33,35,25)(21,26,22,37)(24,30,39,31)(29,44,46,42)(34,41,36,40)(38,45,48,47);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,21,30,23,32,22,31)(18,24,37,43,27,39,26,28)(20,29,36,47,35,46,34,45)(25,38,40,44,33,48,41,42);(1,3,4,8)(2,6,5,7)(11,15)(13,16)(17,20,23,35)(18,25,27,33)(19,29,32,46)(21,36,22,34)(24,38,39,48)(26,41,37,40)(28,42,43,44)(30,47,31,45);(11,15)(13,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,35,36)(24,28,39,43)(25,40,33,41)(29,45,46,47)(38,44,48,42);(10,14)(13,16)(20,35)(25,33)(29,46)(34,36)(38,48)(40,41)(42,44)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,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, 6382690);