#181412

order	512
nilpotency class	3
isoclinism	isoclinic class of #181344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1225 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,12,15)(3,9,11,16,10,6,13,14)(17,18,21,26,23,27,22,28)(19,29,32,38,33,24,31,39)(20,25,35,42,36,41,34,40)(30,43,44,48,45,37,46,47);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,22,32,23,33,21,31)(18,24,28,39,27,29,26,38)(20,30,34,44,36,45,35,46)(25,37,40,47,41,43,42,48);(17,20,23,36)(18,25,27,41)(19,30,33,45)(21,35,22,34)(24,37,29,43)(26,42,28,40)(31,46,32,44)(38,48,39,47);(17,21,23,22)(18,26,27,28)(19,31,33,32)(20,34,36,35)(24,38,29,39)(25,40,41,42)(30,44,45,46)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=hi,g^2=h,b^a=be,e^a=eh,g^a=gi,e^b=ei,f^b=fh,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88608);