#379666

order	512
nilpotency class	3
isoclinism	isoclinic class of #27987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2504
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14249
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

794 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)(3,12)(4,17,20,25)(5,9)(6,11)(7,14)(8,19,26,29)(10,21)(13,28,18,22)(15,24)(16,23)(27,31,32,30);(1,3,5,14,6,16,21,15)(2,7,10,12,11,24,9,23)(4,18,19,31,20,13,29,30)(8,22,25,27,26,28,17,32);(1,4,6,20)(2,8)(3,13)(5,19,21,29)(7,22,24,28)(9,17)(10,25)(11,26)(12,27,23,32)(14,30)(15,31)(16,18);(2,9,11,10)(7,23,24,12)(8,17,26,25)(22,32,28,27)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=ef,d^2=gh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,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, 286862);