#364430

order	512
nilpotency class	3
isoclinism	isoclinic class of #196161
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

2384 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)(5,6,10,16,11,15,9,14)(7,12,18,27,19,28,25,31)(8,20,23,13,24,30,22,29)(17,32,33,36,21,35,34,26);(2,4)(5,7,10,18,11,19,9,25)(6,12,16,27,15,28,14,31)(8,21,23,34,24,17,22,33)(13,26,30,32,29,36,20,35);(5,8,11,24)(6,13,15,29)(7,17,19,21)(9,23,10,22)(12,26,28,36)(14,30,16,20)(18,34,25,33)(27,35,31,32);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,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, 271626);