#6848548

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fh,d^2=i,e^2=fh,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, 6755744);