#248818

order	512
nilpotency class	3
isoclinism	isoclinic class of #248743
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61718	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=hi,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=ghi,c^b=cg,e^b=eh,f^b=fi,d^c=di,e^c=eh,e^d=eh,f^d=fh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156014);