#232523

order	512
nilpotency class	3
isoclinism	isoclinic class of #232515
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	#9980	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1298 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)(9,13)(10,12)(11,14)(15,16)(17,18)(19,29)(20,25)(21,28)(22,27)(23,26)(24,31)(30,45)(32,40)(33,39)(34,48)(35,44)(36,42)(37,43)(38,47)(41,46);(6,12)(13,16)(17,19)(18,24)(20,34)(21,33)(22,31)(23,32)(25,41)(26,39)(27,40)(28,29)(30,35)(36,47)(37,46)(38,43)(42,45)(44,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,42)(27,43)(28,44)(29,41)(31,34)(32,46)(33,47)(39,48)(40,45);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(18,26)(24,39)(25,42)(27,28)(29,40)(38,48)(41,45)(43,44)

Polycyclic presentation:

2^9|d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fh,g^b=gh,e^c=ei,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, 139719);