#117839

order	512
nilpotency class	3
isoclinism	isoclinic class of #117832
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	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

523 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:

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

Polycyclic presentation:

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

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, 25035);