#119511

order	512
nilpotency class	3
isoclinism	isoclinic class of #119510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
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

692 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)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,28,32,24)(20,33,36,43)(21,27)(22,37)(23,26)(25,35,42,38)(29,41)(30,44,31,40)(34,39)(45,47)(46,48);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(17,19)(18,24)(20,34)(21,31)(22,30)(23,32)(25,41)(26,28)(27,44)(29,35)(33,45)(36,47)(37,40)(38,46)(39,43)(42,48);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(17,20)(18,25,26,42)(19,29)(21,35)(22,38)(23,36)(24,39,28,45)(27,33,37,43)(30,34)(31,47)(32,46)(40,48,44,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,f^b=fh,g^b=gh,f^c=fh,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, 26707);