#497402

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3293	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1038 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)(7,21)(8,26)(9,16)(10,19)(11,36)(12,17)(13,20)(14,24)(15,29)(18,23)(22,44)(25,49)(27,59)(28,63)(30,60)(31,50)(32,43)(33,42)(34,48)(35,47)(37,52)(38,41)(39,51)(40,62)(45,65)(46,64)(53,55)(54,58)(56,68)(57,67)(61,66);(5,7)(6,14)(8,22)(9,11)(10,32)(12,24)(13,25)(15,40)(16,47)(17,49)(18,42)(19,21)(20,43)(23,31)(26,58)(27,37)(28,56)(29,55)(30,57)(33,38)(34,35)(36,50)(39,54)(41,48)(44,63)(45,66)(46,59)(51,67)(52,53)(60,68)(61,64)(62,65);(5,8,13,30)(6,15,20,46)(7,22,25,57)(9,27,33,62)(10,28,12,39)(11,37,38,65)(14,40,43,59)(16,44,42,67)(17,45,19,52)(18,51,47,63)(21,53,49,66)(23,29,35,64)(24,54,32,56)(26,41,60,36)(31,55,34,61)(48,68,50,58);(5,9)(6,16)(7,23)(8,27)(10,34)(11,32)(12,31)(13,33)(14,41)(15,44)(17,50)(18,21)(19,48)(20,42)(22,29)(24,38)(25,35)(26,59)(28,61)(30,62)(36,43)(37,56)(39,55)(40,60)(45,58)(46,67)(47,49)(51,53)(52,68)(54,65)(57,64)(63,66)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404598);