#476304

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1236 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)(7,17)(8,13)(9,14)(10,29)(11,16)(12,20)(15,33)(18,48)(19,49)(21,45)(22,57)(23,46)(24,40)(25,60)(26,42)(27,47)(28,44)(30,36)(31,39)(32,43)(34,55)(35,56)(37,51)(38,58)(41,54)(50,65)(52,62)(53,64)(59,66)(61,68)(63,67);(2,4)(5,7,11,21)(6,12,16,36)(8,18,24,52)(9,25,28,56)(10,30,33,20)(13,37,40,34)(14,41,44,49)(15,45,29,17)(19,32,54,27)(22,53,58,68)(23,55,31,51)(26,50,59,67)(35,47,60,43)(38,65,57,63)(39,62,46,48)(42,64,66,61);(1,2)(3,4)(5,8)(6,13)(7,18)(9,26)(10,31)(11,24)(12,34)(14,42)(15,46)(16,40)(17,48)(19,53)(20,55)(21,52)(22,27)(23,33)(25,50)(28,59)(29,39)(30,51)(32,58)(35,63)(36,37)(38,43)(41,61)(44,66)(45,62)(47,57)(49,64)(54,68)(56,67)(60,65);(2,4)(5,9,11,28)(6,14,16,44)(7,19,21,54)(8,22,24,58)(10,32,33,27)(12,35,36,60)(13,38,40,57)(15,47,29,43)(17,49,45,41)(18,50,52,67)(20,56,30,25)(23,59,31,26)(34,61,37,64)(39,66,46,42)(48,63,62,65)(51,68,55,53)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383500);