#502399

order	512
nilpotency class	4
isoclinism	isoclinic class of #502133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,5)(3,7,8,6)(9,10,17,40,37,52,15,22,13,20,14,21,16,23,36,51)(11,24,41,44,62,46,38,45,26,18,27,56,29,57,28,53)(12,30,66,50,39,68,34,19,32,67,33,54,35,49,42,48)(25,58,64,72,65,71,63,55,59,69,60,70,61,47,31,43);(2,5)(6,7)(9,11,14,27,37,62,36,28,13,26,17,41,16,29,15,38)(10,18,21,44,52,57,51,45,20,24,40,56,23,46,22,53)(12,31,33,59,39,64,42,61,32,63,66,25,35,60,34,65)(19,47,30,55,54,58,68,70,48,71,67,43,50,69,49,72);(9,12,13,32)(10,19,20,48)(11,25,26,59)(14,34,17,42)(15,33,36,66)(16,39,37,35)(18,43,24,55)(21,49,40,68)(22,30,51,67)(23,54,52,50)(27,63,41,31)(28,64,38,60)(29,65,62,61)(44,71,56,47)(45,58,53,69)(46,72,57,70);(1,3,4,8)(2,6,5,7)(10,20)(18,24)(19,48)(21,40)(22,51)(23,52)(30,67)(43,55)(44,56)(45,53)(46,57)(47,71)(49,68)(50,54)(58,69)(70,72)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409595);