#456635

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1359 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi

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, 363831);