#398812

order	512
nilpotency class	3
isoclinism	isoclinic class of #398812
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14769
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

869 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:

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

Polycyclic presentation:

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

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, 306008);