#233753

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

1163 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=hi,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, 140949);