#114951

order	512
nilpotency class	3
isoclinism	isoclinic class of #114945
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	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

826 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22147);