#233786

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1183 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,d^c=di,e^c=ei,f^c=fhi

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, 140982);