#4433699

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=ghi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4340895);