#182133

order	512
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

907 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,f^2=i,g^2=h,b^a=be,c^a=ci,g^a=gh,d^b=di,e^b=eh,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, 89329);