#222267

order	512
nilpotency class	3
isoclinism	isoclinic class of #222256
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=hi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,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, 129463);