#118762

order	512
nilpotency class	3
isoclinism	isoclinic class of #117993
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

507 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,f^c=fhi,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, 25958);