#222255

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2,5,9,6,10,4,8)(3,7,13,16,11,14,12,15)(17,18)(19,29)(20,34,37,25)(21,26)(22,41)(23,28)(24,32)(27,44)(30,57,60,55)(31,56)(33,53)(35,64,68,48)(36,49,71,65)(38,72)(39,54)(40,52)(42,47)(43,51)(45,59,63,69)(46,62)(50,61)(58,74,78,77)(66,73,79,80)(67,75,70,76);(1,3)(2,7)(4,12)(5,13)(6,11)(8,15)(9,16)(10,14)(17,19,23,33)(18,24,28,47)(20,30,37,60)(21,38,40,62)(22,42,44,32)(25,45,34,63)(26,50,52,56)(27,53,41,29)(31,43,61,39)(35,66,68,79)(36,69,71,59)(46,54,72,51)(48,74,64,77)(49,57,65,55)(58,70,78,67)(73,76,80,75);(3,11)(7,14)(12,13)(15,16)(17,20,23,37)(18,25,28,34)(19,30,33,60)(21,35,40,68)(22,36,44,71)(24,45,47,63)(26,48,52,64)(27,49,41,65)(29,55,53,57)(31,58,61,78)(32,59,42,69)(38,66,62,79)(39,67,43,70)(46,73,72,80)(50,74,56,77)(51,75,54,76);(1,4,6,5)(2,8,10,9)(3,12,11,13)(7,15,14,16)(17,21,23,40)(18,26,28,52)(19,31,33,61)(20,35,37,68)(22,43,44,39)(24,46,47,72)(25,48,34,64)(27,54,41,51)(29,56,53,50)(30,58,60,78)(32,62,42,38)(36,70,71,67)(45,73,63,80)(49,76,65,75)(55,77,57,74)(59,79,69,66)

Polycyclic presentation:

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

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, 129451);