#233751

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,11,16)(7,17,20,29)(8,21,22,30)(9,15,10,14)(12,24,27,19)(13,25,28,23)(18,26,32,36)(31,33,35,34)(37,38,43,47)(39,48,52,61)(40,45,55,60)(41,46,42,56)(44,50,58,51)(49,64,67,63)(53,59,54,62)(57,65,68,66);(1,2)(3,4)(5,7)(6,12)(8,18)(9,24)(10,19)(11,20)(13,26)(14,29)(15,17)(16,27)(21,35)(22,32)(23,34)(25,33)(28,36)(30,31)(37,39,43,52)(38,44,47,58)(40,49,55,67)(41,51,42,50)(45,57,60,68)(46,61,56,48)(53,66,54,65)(59,63,62,64);(2,4)(5,8,11,22)(6,13,16,28)(7,18,20,32)(9,23,10,25)(12,26,27,36)(14,21,15,30)(17,31,29,35)(19,33,24,34)(37,40,43,55)(38,45,47,60)(39,49,52,67)(41,54,42,53)(44,57,58,68)(46,62,56,59)(48,63,61,64)(50,66,51,65);(8,22)(13,28)(18,32)(21,30)(23,25)(26,36)(31,35)(33,34)(37,41,43,42)(38,46,47,56)(39,50,52,51)(40,53,55,54)(44,48,58,61)(45,59,60,62)(49,65,67,66)(57,64,68,63)

Polycyclic presentation:

2^9|a^2=h,b^2=i,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=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, 140947);