#117247

order	512
nilpotency class	3
isoclinism	isoclinic class of #117239
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

577 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)(19,29,33,39,32,40,31,24)(20,25,35,43,36,44,37,42)(30,45,34,38,46,41,47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,16)(10,14)(17,19,21,31,23,32,22,33)(18,24,26,29,28,39,27,40)(20,34,35,46,36,47,37,30)(25,41,42,38,44,45,43,48);(2,8)(3,11)(4,5)(7,14)(12,13)(15,16)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,38)(26,43)(27,42)(28,44)(29,41)(31,47)(32,46)(33,34)(39,48)(40,45)

Polycyclic presentation:

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

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, 24443);