#273572

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	#61415	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,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, 180768);