#495243

order	512
nilpotency class	4
isoclinism	isoclinic class of #483839
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
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 ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1048 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)(7,20,26,41)(8,27,32,46)(9,16)(10,36)(11,37)(12,38)(13,19)(14,24,42,25)(15,31,47,30)(17,35)(18,21)(22,53)(23,49,34,50)(28,61)(29,59,33,66)(39,65)(40,51,43,57)(44,56)(45,62,60,52)(48,54)(55,67)(58,64)(63,68);(1,2,3,4)(5,7,21,51,9,34,11,24,13,26,35,57,12,23,10,25)(6,14,37,50,38,40,18,20,19,42,36,49,16,43,17,41)(8,28,62,54,29,22,30,64,32,65,52,55,33,56,31,63)(15,44,66,48,60,39,46,68,47,53,59,67,45,61,27,58);(5,8,11,31,12,33,21,52,13,32,10,30,9,29,35,62)(6,15,18,27,16,45,37,59,19,47,17,46,38,60,36,66)(7,22,25,55,23,28,57,64,26,56,24,54,34,65,51,63)(14,39,41,67,43,44,49,68,42,61,20,48,40,53,50,58);(5,9,13,12)(6,16,19,38)(7,23,26,34)(8,29,32,33)(10,21,11,35)(14,40,42,43)(15,45,47,60)(17,36,18,37)(20,49,41,50)(22,28,56,65)(24,51,25,57)(27,59,46,66)(30,52,31,62)(39,53,61,44)(48,68,67,58)(54,63,55,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343344, #72343347, #72343353, …
maximal normal subgroup of	#72343344, #72343347, #72343353, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402439);