#4309108

order	512
nilpotency class	2
isoclinism	isoclinic class of #4308740
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

3514 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df

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, 4216304);