#181907

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄×C₈	C₂³
commutator, socle	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,d^a=di,d^b=dh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89103);