#331607

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

593 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,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, 238803);