#331610

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

Polycyclic presentation:

2^9|a^2=efh,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, 238806);