#439371

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1124 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:

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

Polycyclic presentation:

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

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, 346567);