#233741

order	512
nilpotency class	3
isoclinism	isoclinic class of #233734
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

1167 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)(7,17)(8,20)(9,14)(10,15)(11,16)(12,19)(13,18)(21,22,27,31)(23,32,36,45)(24,29,39,44)(25,30,26,40)(28,34,42,35)(33,48,51,47)(37,43,38,46)(41,49,52,50);(1,2)(3,4)(5,7)(6,12)(8,11)(9,18)(10,19)(13,16)(14,17)(15,20)(21,23)(22,28)(24,33)(25,35)(26,34)(27,36)(29,41)(30,45)(31,42)(32,40)(37,50)(38,49)(39,51)(43,47)(44,52)(46,48);(2,4)(5,8,11,7)(6,13,16,12)(9,18,10,19)(14,17,15,20)(21,24,27,39)(22,29,31,44)(23,33,36,51)(25,38,26,37)(28,41,42,52)(30,46,40,43)(32,47,45,48)(34,50,35,49);(5,9,11,10)(6,14,16,15)(7,18,8,19)(12,17,13,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,39,38)(28,32,42,45)(29,43,44,46)(33,49,51,50)(41,48,52,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140937);