#191107

order	512
nilpotency class	3
isoclinism	isoclinic class of #163899
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1046 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)(11,17)(12,20,21,15)(13,16)(14,19)(18,22,24,23)(25,26)(27,34)(28,32)(29,31)(30,35)(33,37)(36,39)(38,40);(2,5)(6,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27)(26,30)(28,35)(29,36)(31,38)(32,34)(33,39)(37,40);(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24)(26,31)(29,37)(30,38)(32,33)(34,39)(36,40);(1,3,4,7)(2,6,5,8)(27,35)(30,34)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=eh,f^2=i,b^a=be,c^a=cf,f^a=fh,d^b=di,f^c=fi,g^c=ghi

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, 98303);