#478210

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=efi,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385406);