#223168

order	512
nilpotency class	3
isoclinism	isoclinic class of #164127
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(9,10,13,18,14,17,11,15)(12,20,19,24,21,16,22,23)(25,26)(27,34)(28,32)(29,37)(30,33)(31,35)(36,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24);(1,3,4,7)(2,6,5,8)(9,12,14,21)(10,16,17,20)(11,19,13,22)(15,23,18,24)(25,27,30,36)(26,31,33,40)(28,35,29,38)(32,39,37,34);(2,5)(6,8)(9,11,14,13)(10,15,17,18)(12,19,21,22)(16,23,20,24)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130364);