#392553

order	512
nilpotency class	3
isoclinism	isoclinic class of #188761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,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,h^c=hi,h^d=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, 299749);