#349068

order	512
nilpotency class	3
isoclinism	isoclinic class of #349030
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18,23,28)(19,29,33,39)(20,34,36,42)(21,27,22,26)(24,32,40,31)(25,37,41,35)(30,44,46,38)(43,47,48,45);(2,6)(7,15)(8,13)(9,16)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,37,47,36,46,35,45)(25,38,42,48,41,44,34,43);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9);(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,42)(28,41)(29,43)(31,45)(32,47)(33,46)(39,48)(40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,f^c=fi,g^d=gh

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, 256264);