#233760

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

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

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=dhi,e^b=eh,f^b=fi,d^c=dh,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, 140956);