#235134

order	512
nilpotency class	3
isoclinism	isoclinic class of #235117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3186	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1262 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)(5,9)(6,10)(8,12)(13,16)(14,15)(17,18)(19,29)(20,34)(21,27)(22,26)(23,28)(24,31)(25,35)(30,41)(32,40)(33,39)(36,43)(37,42)(38,46)(44,45)(47,48);(2,7)(4,5)(8,15)(9,10)(11,16)(12,13)(17,19,23,33)(18,24,28,40)(20,30,36,46)(21,32,22,31)(25,41,42,38)(26,39,27,29)(34,44,43,48)(35,47,37,45);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,25)(19,30)(21,35)(22,37)(23,36)(24,38)(26,34)(27,43)(28,42)(29,44)(31,45)(32,47)(33,46)(39,48)(40,41);(1,4,6,5)(2,7,10,9)(3,12,14,13)(8,15,16,11)

Polycyclic presentation:

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

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, 142330);