#222271

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

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,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, 129467);