#512744

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₃₂	C₂³
commutator, socle	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#532	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

440 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,f^2=g,g^2=h,h^2=i,b^a=be,d^a=di,d^b=di

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231958
maximal normal subgroup of	#72231958

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419940);