#118585

order	512
nilpotency class	3
isoclinism	isoclinic class of #118582
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	C₂³×D₄	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=ehi,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, 25781);