#498361

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

680 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=ehi,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=de,e^b=eh,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405557);