#114753

order	512
nilpotency class	3
isoclinism	isoclinic class of #114746
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi,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, 21949);