#496129

order	512
nilpotency class	4
isoclinism	isoclinic class of #483962
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

733 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)(5,6,10,15)(7,17)(8,13,23,36)(9,26,27,56)(11,30,29,16)(12,31)(14,39,40,28)(18,41)(19,42)(20,35)(21,50)(22,43)(24,53,51,68)(25,55,52,38)(32,57)(33,48)(34,49)(37,62,61,54)(44,63)(45,59)(46,65)(47,64)(58,66)(60,67);(5,7,9,20,11,21,28,49,10,22,27,31,29,19,39,48)(6,12,14,17,16,34,26,42,15,35,40,43,30,33,56,50)(8,18,24,45,25,44,54,66,23,46,51,57,52,47,62,67)(13,32,37,41,38,58,53,64,36,59,61,65,55,60,68,63);(1,2,3,4)(5,8,10,23)(6,13,15,36)(7,18,22,46)(9,24,27,51)(11,25,29,52)(12,32,35,59)(14,37,40,61)(16,38,30,55)(17,41,43,65)(19,47,21,44)(20,45,31,57)(26,53,56,68)(28,54,39,62)(33,60,34,58)(42,64,50,63)(48,67,49,66);(7,19,22,21)(8,23)(9,27)(12,33,35,34)(13,36)(14,40)(17,42,43,50)(18,44,46,47)(20,49,31,48)(25,52)(26,56)(28,39)(32,58,59,60)(38,55)(41,63,65,64)(45,67,57,66)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343336, #72343356, #72343488
maximal normal subgroup of	#72343336, #72343356, #72343488

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403325);