#6852708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6759904);