#497403

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
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×#3290	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1053 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)(7,18)(8,14)(9,30)(10,33)(11,17)(12,21)(13,27)(15,32)(16,31)(19,37)(20,51)(22,38)(23,40)(24,39)(25,59)(26,47)(28,46)(29,48)(34,42)(35,63)(36,54)(41,66)(43,61)(44,62)(45,55)(49,64)(50,53)(52,57)(56,65)(58,67)(60,68);(5,7)(6,12)(8,25)(9,23)(10,22)(11,24)(13,41)(14,45)(15,39)(16,38)(17,40)(18,30)(19,29)(20,26)(21,32)(27,60)(28,55)(31,57)(33,65)(34,66)(35,58)(36,44)(37,46)(42,50)(43,68)(47,59)(48,51)(49,67)(52,63)(53,62)(54,61)(56,64);(5,8)(6,13)(7,19)(9,26)(10,34)(11,29)(12,36)(14,33)(15,42)(16,47)(17,44)(18,50)(20,21)(22,54)(23,55)(24,25)(27,31)(28,32)(30,43)(35,61)(37,38)(39,68)(40,41)(45,52)(46,49)(48,63)(51,65)(53,57)(56,60)(58,66)(59,67)(62,64);(5,8)(6,14)(7,20)(9,28)(10,27)(11,29)(12,37)(13,33)(15,47)(16,42)(17,48)(18,51)(19,21)(22,54)(23,25)(24,55)(26,32)(30,46)(31,34)(35,62)(36,38)(39,45)(40,59)(41,67)(43,49)(44,63)(50,65)(52,68)(53,56)(57,60)(58,66)(61,64)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404599);