#223079

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130275);