#331635

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
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

540 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)(17,18,21,31,22,25,23,24)(19,26,32,30,28,20,27,29)(33,34)(35,38)(36,40)(37,39)(41,45)(42,47)(43,46)(44,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,20)(21,32)(22,28)(23,27)(24,30)(25,26)(29,31)(33,35,36,42,37,43,44,41)(34,38,40,47,39,46,48,45);(4,12)(7,14)(10,16)(13,15)(17,20,22,26)(18,19,25,28)(21,30,23,29)(24,32,31,27)(35,41)(36,44)(38,45)(40,48)(42,43)(46,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21,22,23)(18,24,25,31)(19,27,28,32)(20,29,26,30)(34,39)(38,46)(40,48)(45,47)

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,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, 238831);