#498822

order	512
nilpotency class	4
isoclinism	isoclinic class of #498821
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1506 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)(7,17)(8,13)(9,25)(10,31)(11,15)(12,19)(14,29)(16,28)(18,45)(20,53)(21,46)(22,37)(23,55)(24,58)(26,43)(27,60)(30,44)(32,40)(33,51)(34,52)(35,62)(36,48)(38,57)(39,54)(41,63)(42,50)(47,64)(49,68)(56,65)(59,67)(61,66);(5,7,11,21)(6,12,15,34)(8,18,22,47)(9,26,30,62)(10,32,28,20)(13,36,37,33)(14,40,42,53)(16,43,31,35)(17,44,46,25)(19,50,52,29)(23,56,27,61)(24,59,54,49)(38,67,41,68)(39,65,58,66)(45,60,64,55)(48,63,51,57);(2,4)(5,8)(6,13)(7,18)(9,27)(10,24)(11,22)(12,33)(14,41)(15,37)(16,39)(17,45)(19,51)(20,49)(21,47)(23,30)(25,60)(26,61)(28,54)(29,63)(31,58)(32,59)(34,36)(35,65)(38,42)(40,67)(43,66)(44,55)(46,64)(48,52)(50,57)(53,68)(56,62);(8,22)(13,37)(18,47)(23,27)(24,54)(33,36)(38,41)(39,58)(45,64)(48,51)(49,59)(55,60)(56,61)(57,63)(65,66)(67,68)

Polycyclic presentation:

2^9|b^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,g^a=gi,h^a=hi,c^b=cg,e^b=eh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281756, #72281854, #72282210
maximal normal subgroup of	#72281756, #72281854, #72282210

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406018);