#464966

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16512
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ 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)(3,13)(4,18)(5,21)(6,25)(7,12)(8,16)(9,20)(10,14)(11,26)(15,29)(17,30)(19,27)(22,28)(23,31)(24,32);(1,3)(2,8,12,28)(4,14)(5,15)(6,16)(7,17)(9,10,27,21)(11,13,25,30)(18,32,29,31)(19,24)(20,23)(22,26);(1,4,6,20,7,15,26,19)(2,9,11,18,12,27,25,29)(3,14,16,23,17,5,22,24)(8,21,13,31,28,10,30,32);(1,5)(2,10)(3,15)(4,17)(6,24)(7,14)(8,27)(9,28)(11,32)(12,21)(13,29)(16,19)(18,30)(20,22)(23,26)(25,31)

Polycyclic presentation:

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

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, 372162);