#503132

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
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	#60649	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1238 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fh,d^2=ei,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,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410328);