#116767

order	512
nilpotency class	3
isoclinism	isoclinic class of #116762
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

554 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,f^b=fi,e^c=ehi,f^c=fi,g^c=gh

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, 23963);