#331624

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=eg,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

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, 238820);