#7622972

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
socle	C₂²	C₂³×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 68 points:

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

Polycyclic presentation:

2^9|b^2=fh,e^2=i,f^2=hi,h^2=i,b^a=bf,c^a=cg,f^a=fh,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343771, #72343834, #72343835
maximal normal subgroup of	#72343771, #72343834, #72343835

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530168);