#117554

order	512
nilpotency class	3
isoclinism	isoclinic class of #10238
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

552 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=i,f^2=i,g^2=hi,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=ei,f^c=fi

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, 24750);