#232557

order	512
nilpotency class	3
isoclinism	isoclinic class of #232546
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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139753);