#6763571

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18,22,27)(19,26,30,23)(20,31,34,40)(21,35,38,25)(24,32,42,28)(29,39,36,43)(33,48,37,47)(41,45,44,46);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,26)(20,32,34,28)(21,36,38,29)(24,40,42,31)(25,43,35,39)(33,45,37,46)(41,47,44,48);(2,6)(3,7)(10,13)(14,16)(17,20,19,28,22,34,30,32)(18,24,23,31,27,42,26,40)(21,37,29,46,38,33,36,45)(25,44,39,48,35,41,43,47);(9,11,12,15)(10,14,13,16)(17,21,22,38)(18,25,27,35)(19,29,30,36)(20,33,34,37)(23,39,26,43)(24,41,42,44)(28,45,32,46)(31,47,40,48);(3,7)(5,8)(17,19,22,30)(18,26,27,23)(20,28,34,32)(21,36,38,29)(24,40,42,31)(25,39,35,43)(33,46,37,45)(41,47,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6670767);