#6911164

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#38948	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1615 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=fi,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=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6818360);