#114560

order	512
nilpotency class	3
isoclinism	isoclinic class of #114551
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

934 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)(3,9,11,6)(4,12,13,7)(5,8)(10,14)(15,16)(17,18,22,27,23,28,21,26)(19,29,32,24,33,39,31,40)(20,25,35,42,36,44,37,43)(30,41,34,38,45,47,46,48);(1,3)(2,6)(4,10)(5,11)(7,15)(8,9)(12,14)(13,16)(17,19,23,33)(18,24,28,40)(20,34,36,46)(21,32,22,31)(25,41,44,47)(26,39,27,29)(30,37,45,35)(38,42,48,43);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,38)(26,43)(27,42)(28,44)(29,41)(31,46)(32,34)(33,45)(39,47)(40,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi,g^c=gh

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, 21756);