#113760

order	512
nilpotency class	3
isoclinism	isoclinic class of #113756
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

773 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)(4,8)(5,7)(10,13,16,12)(14,15)(17,18,22,28,23,27,21,26)(19,29,32,24,33,39,31,40)(20,34,36,25,37,43,35,42)(30,41,46,48,47,38,45,44);(1,3)(2,6)(4,13)(5,11)(7,9)(8,10)(12,15)(14,16)(17,19)(18,24)(20,30)(21,32)(22,31)(23,33)(25,41)(26,39)(27,40)(28,29)(34,48)(35,46)(36,45)(37,47)(38,42)(43,44);(1,2)(3,10)(4,14)(5,7)(6,13)(8,15)(9,12)(11,16)(17,20,23,37)(18,25,27,42)(19,30,33,47)(21,36,22,35)(24,38,40,41)(26,43,28,34)(29,44,39,48)(31,46,32,45)

Polycyclic presentation:

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

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, 20956);