#431997

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

1138 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)(5,6,10,16,12,17,11,33)(7,18,21,38,23,37,22,13)(8,24,27,42,28,41,32,14)(9,15,30,44,31,46,34,45)(19,48,50,47,51,35,25,39)(20,43,29,59,53,61,52,36)(26,54,56,64,58,65,57,40)(49,63,67,66,68,60,55,62);(5,7,10,22,12,23,11,21)(6,13,16,37,17,38,33,18)(8,25,27,51,28,50,32,19)(9,29,30,20,31,52,34,53)(14,39,24,35,42,47,41,48)(15,43,44,36,46,61,45,59)(26,49,56,55,58,68,57,67)(40,63,54,62,64,60,65,66);(5,8,12,28)(6,14,17,42)(7,19)(9,26,31,58)(10,32,11,27)(13,35)(15,40,46,64)(16,41,33,24)(18,47)(20,49)(21,25)(22,50)(23,51)(29,55)(30,57,34,56)(36,60)(37,39)(38,48)(43,66)(44,65,45,54)(52,67)(53,68)(59,63)(61,62);(2,4)(5,9)(6,15)(7,20,23,53)(8,26)(10,30)(11,34)(12,31)(13,36,38,59)(14,40)(16,44)(17,46)(18,43,37,61)(19,49,51,68)(21,29,22,52)(24,54)(25,55,50,67)(27,56)(28,58)(32,57)(33,45)(35,60,48,63)(39,62,47,66)(41,65)(42,64)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ef,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^d=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, 339193);