#364469

order	512
nilpotency class	3
isoclinism	isoclinic class of #196134
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2394 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)(3,4)(5,6,10,15,11,14,9,25)(7,12,19,27,17,28,18,34)(8,20,23,31,24,13,22,30)(16,26,33,29,32,36,21,35);(2,4)(5,7,10,19,11,17,9,18)(6,12,15,27,14,28,25,34)(8,21,23,16,24,33,22,32)(13,29,30,36,20,35,31,26);(5,8,11,24)(6,13,14,20)(7,16,17,32)(9,23,10,22)(12,26,28,36)(15,31,25,30)(18,33,19,21)(27,35,34,29);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,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, 271665);