#360371

order	512
nilpotency class	3
isoclinism	isoclinic class of #360370
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13874
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1101 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,f^c=fi,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, 267567);