#191237

order	512
nilpotency class	3
isoclinism	isoclinic class of #191234
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

551 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)(19,27,32,23)(20,33,36,41)(21,25)(22,26)(24,35,42,34)(28,37)(29,44)(30,40,31,39)(38,45)(43,47)(46,48);(1,3,13,14,6,15,5,4)(2,7,9,8,11,16,10,12)(17,19,28,30,22,32,21,31)(18,23,37,39,26,27,25,40)(20,29,34,47,36,46,35,45)(24,38,41,44,42,43,33,48);(1,4,5,15,6,14,13,3)(2,8,10,7,11,12,9,16)(17,20)(18,24,26,42)(19,29)(21,35)(22,36)(23,38,27,43)(25,33,37,41)(28,34)(30,47)(31,45)(32,46)(39,44,40,48);(1,5,6,13)(2,9,11,10)(3,14,15,4)(7,12,16,8)(17,21,22,28)(18,25,26,37)(19,30,32,31)(20,34,36,35)(23,39,27,40)(24,41,42,33)(29,45,46,47)(38,48,43,44)

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=dh,g^b=ghi,d^c=dh,g^c=ghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98433);