#117810

order	512
nilpotency class	3
isoclinism	isoclinic class of #117805
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

569 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)(3,7)(4,6)(5,12)(8,11)(9,15)(10,14)(13,16)(17,18,22,26,23,28,21,27)(19,29,32,39,33,43,31,24)(20,34,35,25,37,42,36,41)(30,38,45,48,46,40,47,44);(2,6)(3,9,10,13)(7,15,14,16)(8,12)(17,19,21,31,23,33,22,32)(18,24,26,29,28,39,27,43)(20,30,35,45,37,46,36,47)(25,40,34,48,41,38,42,44);(1,3)(2,7)(4,10)(5,13)(6,14)(8,15)(9,11)(12,16)(17,20,23,37)(18,25,28,41)(19,30,33,46)(21,35,22,36)(24,38,39,40)(26,34,27,42)(29,44,43,48)(31,45,32,47)

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,e^b=eh,d^c=di,e^c=eh,f^c=fi,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, 25006);