#191231

order	512
nilpotency class	3
isoclinism	isoclinic class of #191231
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

532 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98427);