#6745773

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2308 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)(19,24)(20,32)(21,26)(22,35)(23,27)(25,33)(28,42)(29,38)(30,39)(31,45)(34,40)(36,41)(37,44)(43,48)(46,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,29)(18,24,27,38)(20,28,34,43)(21,30,22,31)(25,37,41,47)(26,39,35,45)(32,42,40,48)(33,44,36,46);(11,15)(13,16)(17,20)(18,25)(19,28)(21,33)(22,36)(23,34)(24,37)(26,40)(27,41)(29,43)(30,44)(31,46)(32,35)(38,47)(39,48)(42,45);(2,6)(5,7)(19,29)(24,38)(28,43)(30,31)(37,47)(39,45)(42,48)(44,46);(3,8)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,35)(19,30,29,31)(20,33,34,36)(24,39,38,45)(25,40,41,32)(28,44,43,46)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,e^2=i,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

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, 6652969);