#273817

order	512
nilpotency class	3
isoclinism	isoclinic class of #273755
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	#62238	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=hi,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fh,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181013);