#205974

order	512
nilpotency class	3
isoclinism	isoclinic class of #205785
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,g^b=gi,d^c=di

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, 113170);