#494899

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
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×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

762 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402095);