#273629

order	512
nilpotency class	3
isoclinism	isoclinic class of #273563
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	C₂×C₄²	C₂⁴
prime radical	#3036	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

815 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:

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

Polycyclic presentation:

2^9|a^2=fi,d^2=i,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,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, 180825);