#114255

order	512
nilpotency class	3
isoclinism	isoclinic class of #114249
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

812 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)(3,9,12,6)(4,8)(5,7)(10,15,11,14)(13,16)(17,18)(19,28,32,41)(20,33)(21,26)(22,37)(23,27)(24,30,39,31)(25,35)(29,38,46,48)(34,40)(36,42)(43,44,47,45);(1,3,5,12)(2,6,7,9)(4,10,13,11)(8,14,16,15)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,38)(26,41)(27,39)(28,37)(33,43)(34,45)(35,44)(36,46)(40,48)(42,47);(3,10)(4,13)(6,14)(8,16)(9,15)(11,12)(17,20,23,36)(18,25,27,40)(19,29,32,46)(21,35,22,34)(24,38,39,48)(26,42,37,33)(28,43,41,47)(30,45,31,44)

Polycyclic presentation:

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

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, 21451);