#116393

order	512
nilpotency class	3
isoclinism	isoclinic class of #116392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
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

563 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)(3,9,11,16)(4,7)(5,8)(6,10,14,13)(12,15)(17,18,21,26,22,25,35,40)(19,23,29,38,28,37,44,47)(20,30,33,41,32,24,34,39)(27,42,43,48,31,36,45,46);(4,12)(6,14)(7,15)(10,13)(17,19)(18,23)(20,31,32,27)(21,29)(22,28)(24,36,30,42)(25,37)(26,38)(33,45,34,43)(35,44)(39,46,41,48)(40,47);(1,3)(2,6)(4,13)(5,11)(7,16)(8,14)(9,15)(10,12)(17,20)(18,24)(19,27)(21,34)(22,32)(23,36)(25,30)(26,41)(28,31)(29,45)(33,35)(37,42)(38,48)(39,40)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,e^c=ei,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, 23589);