#215640

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1068 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)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26)(27,30)(28,35,36,31)(29,32)(33,39,40,37)(34,38);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(2,5)(3,7)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39);(10,16)(14,17)(15,24)(22,23)(27,34)(30,38)(33,40)(37,39)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122836);