#202018

order	512
nilpotency class	3
isoclinism	isoclinic class of #201938
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,f^b=fi,d^c=dh,e^c=ei,f^c=fh,g^c=gi

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, 109214);