#497852

order	512
nilpotency class	4
isoclinism	isoclinic class of #33856
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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×#135	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

598 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)(11,37)(12,17)(13,20)(14,24)(15,29)(18,23)(22,52)(25,49)(27,51)(28,58)(30,57)(31,59)(32,50)(33,42)(34,41)(35,48)(36,47)(38,40)(39,60)(43,61)(44,53)(45,64)(46,65)(54,56)(55,67)(62,68)(63,66);(5,7)(6,14)(8,22)(9,11)(10,33)(12,24)(13,25)(15,39)(16,47)(17,49)(18,41)(19,21)(20,42)(23,32)(26,56)(27,29)(28,61)(30,54)(31,55)(34,38)(35,36)(37,50)(40,48)(43,59)(44,68)(45,51)(46,66)(52,58)(53,60)(57,67)(62,65)(63,64);(5,8,23,53,12,30,38,65,13,31,36,64,10,28,11,29)(6,15,37,58,19,45,47,59,20,46,40,57,17,44,18,26)(7,22,32,60,24,54,34,62,25,55,35,63,33,61,9,27)(14,39,50,52,21,51,16,43,42,66,48,67,49,68,41,56);(5,9)(6,16)(7,23)(8,27)(10,35)(11,33)(12,32)(13,34)(14,40)(15,43)(17,50)(18,21)(19,48)(20,41)(22,53)(24,38)(25,36)(26,51)(28,63)(29,61)(30,60)(31,62)(37,42)(39,57)(44,52)(45,67)(46,56)(47,49)(54,65)(55,64)(58,66)(59,68)

Polycyclic presentation:

2^9|a^2=g,c^2=fh,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,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, 405048);