#118412

order	512
nilpotency class	3
isoclinism	isoclinic class of #118411
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	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

701 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)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18)(19,29)(20,33,36,25)(21,28)(22,26)(23,27)(24,31)(30,45,47,41)(32,43)(34,48,46,39)(35,42,38,44)(37,40);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19,21,31,23,32,22,37)(18,24,26,29,27,40,28,43)(20,34,35,30,36,46,38,47)(25,41,42,39,33,45,44,48);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(17,20,23,36)(18,25,27,33)(19,30,32,47)(21,35,22,38)(24,39,40,48)(26,42,28,44)(29,45,43,41)(31,46,37,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25608);