#197556

order	512
nilpotency class	3
isoclinism	isoclinic class of #197348
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1096 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=h,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gh

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, 104752);