#274006

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^2=hi,e^2=i,f^2=hi,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=fhi,g^c=gh,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181202);