#207757

order	512
nilpotency class	3
isoclinism	isoclinic class of #196002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

991 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)(3,6)(4,5)(7,8)(9,10)(11,14,18,22)(12,19)(13,16)(15,20)(17,23,24,21)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(9,11,13,18)(10,14)(12,17,20,24)(15,21)(16,22)(19,23)(25,27,28,34,29,33,35,39)(26,30,31,37,32,36,38,40);(2,5)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23);(1,3,4,7)(2,6,5,8)(12,20)(15,19)(17,24)(21,23)(27,33)(30,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=fh,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=eh,g^b=gh,d^c=dh

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, 114953);