#6791955

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6699151);