#273782

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	#62092	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)(3,8,11,15)(4,7,13,14)(5,9)(6,10)(12,16)(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);(1,3,6,11)(2,7,10,14)(4,12,13,5)(8,9,15,16)(17,19)(18,25)(20,37)(21,33)(22,41)(23,34)(24,35)(26,46)(27,43)(28,44)(29,31)(30,45)(32,39)(36,47)(38,48)(40,42);(3,11)(5,12)(7,14)(9,16)(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);(1,4)(2,8)(3,12)(5,11)(6,13)(7,9)(10,15)(14,16)(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=hi,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, 180978);