#392541

order	512
nilpotency class	3
isoclinism	isoclinic class of #188761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

808 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)(11,19)(12,26,30,35)(13,17)(14,21)(15,20)(16,22)(18,29,36,28)(23,37,39,31)(24,33)(25,32)(27,34,38,40);(2,6)(7,8)(9,11,15,24,16,25,14,13)(10,17,21,32,22,33,20,19)(12,23,29,38,30,39,28,27)(18,31,35,40,36,37,26,34);(1,3,5,4)(2,7,6,8)(9,12)(10,18,22,36)(11,23)(13,27)(14,28)(15,29)(16,30)(17,31,33,37)(19,34,32,40)(20,26,21,35)(24,38)(25,39);(1,4,5,3)(2,8,6,7)(9,13,14,25,16,24,15,11)(10,19,20,33,22,32,21,17)(12,27,28,39,30,38,29,23)(18,34,26,37,36,40,35,31)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299737);