#456896

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

1139 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,e^d=ei,f^d=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, 364092);