#361249

order	512
nilpotency class	3
isoclinism	isoclinic class of #360226
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2352
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13893
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

691 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)(4,18,22,27)(5,10)(6,11)(7,12)(8,15)(9,21,28,20)(14,29,31,23)(16,25)(17,24)(19,32,30,26);(1,3,5,15,7,17,6,16)(2,8,11,13,12,25,10,24)(4,19,20,14,22,30,21,31)(9,26,18,29,28,32,27,23);(1,4,5,20,7,22,6,21)(2,9,11,18,12,28,10,27)(3,14,15,30,17,31,16,19)(8,23,13,26,25,29,24,32);(2,10,12,11)(8,24,25,13)(9,27,28,18)(23,32,29,26)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268445);