#181369

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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1239 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:

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

Polycyclic presentation:

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

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, 88565);