#113540

order	512
nilpotency class	3
isoclinism	isoclinic class of #97710
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

797 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20736);