#117386

order	512
nilpotency class	3
isoclinism	isoclinic class of #117381
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24582);