#496475

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60682	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403671);