#7622936

order	512
nilpotency class	4
isoclinism	isoclinic class of #7622820
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
prime radical	#60643	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2028 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343750, #72343835
maximal normal subgroup of	#72343750, #72343835

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530132);