#6745804

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×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6653000);