#202492

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂×#388
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

663 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,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, 109688);