#202002

order	512
nilpotency class	3
isoclinism	isoclinic class of #201918
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

1232 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)(17,18,22,27,23,28,21,26)(19,29,31,39,33,42,32,24)(20,34,36,25,37,40,35,41)(30,38,44,43,46,48,45,47);(1,3,5,10)(2,6,8,14)(4,9,12,15)(7,13,11,16)(17,19)(18,24,28,39)(20,30)(21,32)(22,31)(23,33)(25,38,41,48)(26,42,27,29)(34,47,40,43)(35,45)(36,44)(37,46);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,15)(14,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,34)(27,40)(28,41)(29,43)(31,44)(32,45)(33,46)(39,48)(42,47);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,42)(25,34,41,40)(30,44,46,45)(38,43,48,47)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=h,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,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, 109198);