#110485

order	512
nilpotency class	3
isoclinism	isoclinic class of #110476
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

582 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efi,e^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17681);