#369875

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,11,18,12,16,10,17)(7,13,23,31,21,32,22,30)(8,24,27,14,26,34,9,15)(19,29,25,28,20,33,35,36);(1,3,4,2)(5,7,10,23,12,21,11,22)(6,13,18,30,16,32,17,31)(8,25,27,19,26,35,9,20)(14,33,24,36,15,29,34,28);(5,8,10,27,12,26,11,9)(6,14,17,34,16,15,18,24)(7,19,22,25,21,20,23,35)(13,28,30,33,32,36,31,29);(5,9,11,26,12,27,10,8)(6,15,17,24,16,14,18,34)(7,20,22,35,21,19,23,25)(13,29,31,36,32,33,30,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=g,d^2=efi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=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, 277071);