#380132

order	512
nilpotency class	3
isoclinism	isoclinic class of #380130
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14259
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1145 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287328);