#331612

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

Polycyclic presentation:

2^9|a^2=efh,b^2=egh,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, 238808);