#273912

order	512
nilpotency class	3
isoclinism	isoclinic class of #273819
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	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=hi,d^2=h,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 181108);