#221200

order	512
nilpotency class	3
isoclinism	isoclinic class of #221184
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	#61284	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

889 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 128396);