#369247

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,8)(5,6)(9,10,13,17,14,18,24,35)(11,15,21,31,22,30,20,29)(12,23,27,16,26,33,25,34)(19,36,37,32,38,40,39,28);(2,6)(7,8)(9,11,13,21,14,22,24,20)(10,15,17,31,18,30,35,29)(12,19,27,37,26,38,25,39)(16,32,33,40,34,28,23,36);(1,3,5,4)(2,7,6,8)(9,12,24,27,14,26,13,25)(10,16,17,23,18,34,35,33)(11,19,20,37,22,38,21,39)(15,28,31,40,30,32,29,36);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=efg,d^2=e,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,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, 276443);