#273766

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	#62089	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 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)(3,10,12,15)(4,8)(5,9)(6,11,7,13)(14,16)(17,18,23,28,24,30,22,29)(19,31,34,25,35,44,41,45)(20,36,38,27,21,40,39,26)(32,47,37,43,48,42,33,46);(1,3)(2,6)(4,13)(5,12)(7,9)(8,15)(10,16)(11,14)(17,19,24,35)(18,25,30,45)(20,37,21,33)(22,41,23,34)(26,46,27,43)(28,44,29,31)(32,38,48,39)(36,47,40,42);(4,14)(8,16)(10,15)(11,13)(17,20,24,21)(18,26,30,27)(19,32,35,48)(22,38,23,39)(25,42,45,47)(28,40,29,36)(31,46,44,43)(33,41,37,34);(1,3)(2,7)(4,13)(5,12)(6,9)(8,10)(11,14)(15,16)(17,21,24,20)(18,27,30,26)(19,33,35,37)(22,38,23,39)(25,43,45,46)(28,40,29,36)(31,47,44,42)(32,41,48,34)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,d^2=h,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, 180962);