#369300

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
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

1270 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)(9,10,14,19,15,18,13,28)(11,16,23,31,21,32,22,37)(12,24,26,34,27,17,29,35)(20,30,25,39,36,40,38,33);(2,6)(7,8)(9,11,14,23,15,21,13,22)(10,16,19,31,18,32,28,37)(12,25,26,36,27,38,29,20)(17,33,35,30,24,39,34,40);(1,3,5,4)(2,7,6,8)(9,12,13,26,15,27,14,29)(10,17,28,35,18,24,19,34)(11,20,23,38,21,36,22,25)(16,30,31,33,32,40,37,39);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=efg,d^2=e,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,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, 276496);