#273414

order	512
nilpotency class	3
isoclinism	isoclinic class of #273307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3494	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

828 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)(3,11,13,7)(4,12)(5,15,16,9)(6,10)(8,14)(17,18)(19,24)(20,32)(21,23)(22,27)(25,33)(26,36)(28,39)(29,44)(30,37)(31,40)(34,41)(35,42)(38,43)(45,47)(46,48);(1,3)(2,7)(4,9)(5,12)(6,13)(8,16)(10,11)(14,15)(18,23)(19,28)(21,36)(25,32)(26,27)(29,43)(30,31)(33,34)(37,39)(41,42)(44,47)(45,46);(1,4)(2,8)(3,9)(5,11)(6,14)(7,16)(10,12)(13,15)(17,19,22,31)(18,24,27,40)(20,29,35,45)(21,30,36,28)(23,37,26,39)(25,38,41,48)(32,44,42,47)(33,43,34,46);(1,5,6,16)(2,9,10,15)(3,12,13,8)(4,11,14,7)(17,20)(18,25)(19,29)(21,34)(22,35)(23,32)(24,38)(26,42)(27,41)(28,43)(30,46)(31,45)(33,36)(37,44)(39,47)(40,48)

Polycyclic presentation:

2^9|a^2=fhi,c^2=i,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=gi,e^d=ei,f^d=fhi

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, 180610);