#235022

order	512
nilpotency class	3
isoclinism	isoclinic class of #235012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2539	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1250 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dhi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142218);