#397947

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1800 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:

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

Polycyclic presentation:

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

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, 305143);