#117312

order	512
nilpotency class	3
isoclinism	isoclinic class of #117305
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

566 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 24508);