#378032

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=fhi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285228);