#117614

order	512
nilpotency class	3
isoclinism	isoclinic class of #117607
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

598 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,f^c=fh,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, 24810);