#6788243

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1342 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)(3,6)(4,5)(7,8)(17,18,22,26)(19,27,30,23)(20,31,34,39)(21,35,37,25)(24,32,41,28)(29,38,46,43)(33,48,36,47)(40,44,42,45);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,29,37,46)(24,39,41,31)(25,38,35,43)(33,45,36,44)(40,48,42,47);(2,5)(6,8)(10,13)(14,16)(17,20,19,28,22,34,30,32)(18,24,23,31,26,41,27,39)(21,36,29,45,37,33,46,44)(25,42,38,48,35,40,43,47);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,29)(20,33)(22,37)(23,38)(24,40)(26,35)(27,43)(28,44)(30,46)(31,47)(32,45)(34,36)(39,48)(41,42);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(21,37)(25,35)(29,46)(33,36)(38,43)(40,42)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6695439);