#232500

order	512
nilpotency class	3
isoclinism	isoclinic class of #232499
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	#60752	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1359 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)(19,28)(20,33)(21,38)(22,27)(23,26)(24,31)(25,36)(29,46)(30,40)(32,41)(34,45)(35,43)(37,44)(39,47)(42,48);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,34)(21,32)(22,31)(23,30)(25,42)(26,41)(27,40)(28,38)(29,35)(33,39)(36,47)(37,48)(43,45)(44,46);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20,23,37)(18,25,26,43)(19,29,30,47)(21,35,22,36)(24,39,41,46)(27,44,38,33)(28,45,40,42)(31,34,32,48);(3,10)(6,13)(9,16)(12,14)(18,26)(19,30)(25,43)(27,38)(29,47)(31,32)(33,44)(34,48)

Polycyclic presentation:

2^9|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=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

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, 139696);