#348176

order	512
nilpotency class	3
isoclinism	isoclinic class of #348158
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3459	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

904 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)(3,7)(4,8)(5,6)(9,15)(10,12)(11,13)(14,16)(17,18,23,28)(19,29,33,40)(20,34,37,44)(21,27,22,26)(24,32,41,31)(25,38,43,36)(30,42,35,39)(45,47,48,46);(2,6)(7,12)(8,13)(14,15)(17,19,22,32,23,33,21,31)(18,24,27,40,28,41,26,29)(20,35,38,46,37,30,36,47)(25,42,44,45,43,39,34,48);(1,3)(2,7)(4,9)(5,10)(6,12)(8,14)(11,16)(13,15)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,34)(27,44)(28,43)(29,45)(31,46)(32,47)(33,35)(40,48)(41,42);(1,4)(2,8)(3,9)(5,11)(6,13)(7,14)(10,16)(12,15)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,c^b=ch,f^c=fh,g^c=gi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255372);