#232657

order	512
nilpotency class	3
isoclinism	isoclinic class of #232654
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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1354 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,32)(25,37)(30,39)(31,40)(33,41)(35,48)(36,43)(38,44)(42,46)(45,47);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19,23,31)(18,24,26,41)(20,35,38,46)(21,33,22,32)(25,42,43,48)(27,40,28,29)(30,37,45,36)(34,47,44,39);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20,23,38)(18,25,26,43)(19,30,31,45)(21,36,22,37)(24,39,41,47)(27,44,28,34)(29,42,40,48)(32,35,33,46);(3,10)(6,13)(9,16)(12,14)(18,26)(19,31)(25,43)(27,28)(30,45)(32,33)(34,44)(35,46)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139853);