#227184

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2358
prime radical	C₂×2₋¹⁺⁴	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

725 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dei,e^b=eh,f^c=fh,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, 134380);