#6753276

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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)(17,18,22,26)(19,23,28,37)(20,30,31,40)(21,33,35,25)(24,34,39,32)(27,41,42,48)(29,45,44,38)(36,46,47,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,28)(18,23,26,37)(20,27,31,42)(21,29,35,44)(24,36,39,47)(25,38,33,45)(30,41,40,48)(32,43,34,46);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,40)(26,39)(28,42)(29,46)(30,33)(32,35)(37,47)(38,48)(41,45)(43,44);(2,6)(5,7)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46);(10,14)(13,15)(20,31)(24,39)(27,42)(30,40)(32,34)(36,47)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,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, 6660472);