#496422

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

887 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,g^c=gh,g^f=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, 403618);