#493080

order	512
nilpotency class	4
isoclinism	isoclinic class of #33469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1102 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)(7,20,24,41)(8,15)(9,16)(10,35)(11,36)(12,37)(13,19)(14,23,40,25)(17,34)(18,33)(21,48,53,66)(22,49,26,58)(27,59,55,68)(28,43)(29,63)(30,64)(31,47)(32,46)(38,54,67,52)(39,56,50,57)(42,60,65,61)(44,51)(45,62);(2,4)(5,7,11,25,12,26,33,56,13,24,10,23,9,22,34,57)(6,14,18,41,16,39,36,58,19,40,17,20,37,50,35,49)(8,27,30,61,28,53,62,52,31,55,29,60,32,21,51,54)(15,42,45,68,46,67,64,66,47,65,44,59,43,38,63,48);(5,8,13,31)(6,15,19,47)(7,21)(9,32,12,28)(10,30,11,29)(14,38)(16,46,37,43)(17,45,18,44)(20,48)(22,55)(23,54)(24,53)(25,52)(26,27)(33,51,34,62)(35,64,36,63)(39,42)(40,67)(41,66)(49,68)(50,65)(56,60)(57,61)(58,59);(5,9,13,12)(6,16,19,37)(7,22,24,26)(8,28,31,32)(10,33,11,34)(14,39,40,50)(15,43,47,46)(17,35,18,36)(20,49,41,58)(21,27,53,55)(23,56,25,57)(29,51,30,62)(38,65,67,42)(44,64,45,63)(48,59,66,68)(52,60,54,61)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fi,d^2=i,e^2=h,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,c^b=ch,f^b=fi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400276);