#360527

order	512
nilpotency class	3
isoclinism	isoclinic class of #360526
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13877
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2215 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267723);