#400996

order	512
nilpotency class	3
isoclinism	isoclinic class of #394044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
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

1656 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308192);