#211083

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

598 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=gh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=dh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118279);