#495140

order	512
nilpotency class	4
isoclinism	isoclinic class of #481349
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1254 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343350, #72343356, #72343362, …
maximal normal subgroup of	#72343350, #72343356, #72343362, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402336);