#191352

order	512
nilpotency class	3
isoclinism	isoclinic class of #188768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

419 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=dh,g^b=gh,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, 98548);