#502908

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33719	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=ei,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,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, 410104);