#233516

order	512
nilpotency class	3
isoclinism	isoclinic class of #233501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6,10,12)(7,15,17,19)(8,14,9,13)(11,18,20,16)(21,22,27,32)(23,33,35,44)(24,38,41,45)(25,31,26,30)(28,37,43,36)(29,39,46,40)(34,48,49,42)(47,50,52,51);(2,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,19)(14,15)(21,23,27,35)(22,28,32,43)(24,34,41,49)(25,37,26,36)(29,42,46,48)(30,44,31,33)(38,52,45,47)(39,51,40,50);(7,16)(8,9)(11,15)(13,14)(17,18)(19,20)(21,24,27,41)(22,29,32,46)(23,34,35,49)(25,40,26,39)(28,42,43,48)(30,38,31,45)(33,47,44,52)(36,51,37,50);(6,12)(7,17)(13,14)(16,18)(23,35)(28,43)(33,44)(34,49)(36,37)(42,48)(47,52)(50,51)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140712);