#495221

order	512
nilpotency class	4
isoclinism	isoclinic class of #33605
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402417);