#331609

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
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

546 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)(17,18,21,25,22,20,23,26)(19,27,30,31,29,24,28,32)(33,34,38,43,36,41,37,42)(35,44,47,40,45,39,46,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,27)(21,30)(22,29)(23,28)(25,32)(26,31)(33,35,37,46,36,45,38,47)(34,39,42,40,41,44,43,48);(4,12)(7,14)(10,16)(13,15)(17,20,22,18)(19,24,29,27)(21,25,23,26)(28,32,30,31)(33,35,36,45)(34,40,41,48)(37,47,38,46)(39,43,44,42);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,22,23)(18,25,20,26)(19,28,29,30)(24,31,27,32)(35,45)(39,44)(40,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238805);