#455945

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1111 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:

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

Polycyclic presentation:

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

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, 363141);