#368891

order	512
nilpotency class	3
isoclinism	isoclinic class of #368888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14023
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1655 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:

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

Polycyclic presentation:

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

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, 276087);