#120914

order	512
nilpotency class	3
isoclinism	isoclinic class of #120912
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=di,f^b=fh,g^b=ghi,e^c=ei,f^c=fh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28110);