#503125

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60648	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1224 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410321);