#120871

order	512
nilpotency class	3
isoclinism	isoclinic class of #120868
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

374 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)(3,11)(4,5)(6,9)(7,10)(8,13)(12,14)(15,16)(17,18,21,26,22,25,36,42)(19,27,30,44,29,23,37,39)(20,31,34,43,33,24,35,41)(28,38,45,47,32,40,46,48);(1,3,6,14,7,15,5,13)(2,8,4,16,10,12,9,11)(17,19)(18,23,25,27)(20,32)(21,30)(22,29)(24,40,31,38)(26,39,42,44)(28,33)(34,46)(35,45)(36,37)(41,48,43,47);(1,4,5,2,7,9,6,10)(3,12,13,16,15,8,14,11)(17,20)(18,24)(19,28,29,32)(21,35)(22,33)(23,38,27,40)(25,31)(26,43)(30,46,37,45)(34,36)(39,48,44,47)(41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^b=fh,f^c=fhi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28067);