#181411

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

1227 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)(17,18,21,27,23,26,22,28)(19,29,31,42,32,24,36,38)(20,25,34,40,33,39,35,41)(30,43,45,48,44,37,46,47);(1,3)(2,6,8,9)(4,11)(5,10)(7,14,15,16)(12,13)(17,19,22,31,23,32,21,36)(18,24,28,38,26,29,27,42)(20,30,35,45,33,44,34,46)(25,37,41,47,39,43,40,48);(17,20)(18,25)(19,30)(21,34)(22,35)(23,33)(24,37)(26,39)(27,40)(28,41)(29,43)(31,45)(32,44)(36,46)(38,47)(42,48);(20,33)(25,39)(30,44)(34,35)(37,43)(40,41)(45,46)(47,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,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, 88607);