#181844

order	512
nilpotency class	2
isoclinism	isoclinic class of #180873
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#376
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

611 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,e^2=h,f^2=i,b^a=be,g^a=gh,d^b=di,f^b=fh,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89040);