#367908

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1123 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=i,d^2=efi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 275104);