#191249

order	512
nilpotency class	3
isoclinism	isoclinic class of #191246
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 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)(3,12)(4,7)(5,9)(6,11)(8,15)(10,13)(14,16)(17,18)(19,27,31,23)(20,32,35,39)(21,25)(22,26)(24,34,40,33)(28,36)(29,44)(30,42,43,38)(37,46)(41,45)(47,48);(1,3,13,14,6,15,5,4)(2,7,9,8,11,16,10,12)(17,19,28,43,22,31,21,30)(18,23,36,42,26,27,25,38)(20,29,33,45,35,47,34,46)(24,37,39,44,40,41,32,48);(1,4,5,15,6,14,13,3)(2,8,10,7,11,12,9,16)(17,20)(18,24,26,40)(19,29)(21,34)(22,35)(23,37,27,41)(25,32,36,39)(28,33)(30,46)(31,47)(38,48,42,44)(43,45);(1,5,6,13)(2,9,11,10)(3,14,15,4)(7,12,16,8)(17,21,22,28)(18,25,26,36)(19,30,31,43)(20,33,35,34)(23,38,27,42)(24,39,40,32)(29,45,47,46)(37,44,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,g^b=ghi,d^c=dh,g^c=ghi

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, 98445);