#118358

order	512
nilpotency class	3
isoclinism	isoclinic class of #118356
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	C₂³×D₄	Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

689 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)(17,18,22,26,23,27,21,36)(19,28,32,39,31,44,30,24)(20,33,34,41,35,25,37,42)(29,43,45,38,46,48,47,40);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19,21,30,23,31,22,32)(18,24,26,28,27,39,36,44)(20,29,34,45,35,46,37,47)(25,40,41,48,33,38,42,43);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,41)(27,33)(28,43)(30,45)(31,46)(32,47)(36,42)(39,40)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,g^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=fi,d^c=di,g^c=ghi

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, 25554);