#405030

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14908
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

656 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312226);