#6848622

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1640 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)(20,35)(21,27)(22,28)(23,39)(24,33)(25,36)(26,37)(30,45)(31,41)(32,46)(34,42)(38,44)(40,48)(43,47);(1,3,4,8)(2,5,6,7)(11,12)(14,15)(17,19)(18,24)(20,31)(21,32)(22,33)(23,34)(25,43)(26,41)(27,42)(28,29)(30,37)(35,45)(36,48)(38,47)(39,46)(40,44);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,40)(26,27)(28,44)(29,43)(31,33)(32,47)(34,48)(35,39)(41,46)(42,45);(2,6)(5,7)(17,20)(18,26)(19,31)(21,38)(22,37)(23,36)(24,41)(25,27)(28,35)(29,45)(30,33)(32,47)(34,48)(39,44)(40,46)(42,43);(9,12,13,11)(10,15,16,14)(17,21,22,23)(18,27,28,39)(19,32,33,34)(20,36,37,38)(24,42,29,46)(25,26,44,35)(30,47,31,48)(40,45,43,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fh,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, 6755818);