#221303

order	512
nilpotency class	3
isoclinism	isoclinic class of #221204
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128499);