#115287

order	512
nilpotency class	3
isoclinism	isoclinic class of #115274
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×#121	C₂×C₄
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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh

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, 22483);