#117965

order	512
nilpotency class	3
isoclinism	isoclinic class of #117963
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×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

553 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)(3,9,11,6)(4,12)(5,7)(8,14)(10,15,13,16)(17,18,22,26,23,27,21,28)(19,29,32,40,33,44,31,24)(20,34,36,42,38,25,37,43)(30,41,35,48,46,45,47,39);(3,10,11,13)(6,15,9,16)(17,19,21,31,23,33,22,32)(18,24,26,29,27,40,28,44)(20,35,36,46,38,47,37,30)(25,41,42,39,34,45,43,48);(1,3)(2,6)(4,13)(5,11)(7,9)(8,16)(10,14)(12,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,42)(27,34)(28,43)(29,45)(31,35)(32,47)(33,46)(40,48)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25161);