#4312906

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3356 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)(6,8)(11,15)(14,16)(17,18,22,26)(19,20,24,23)(21,29,31,25)(27,32,30,28);(10,13)(14,16)(17,19,22,24)(18,23,26,20)(21,30,31,27)(25,28,29,32);(5,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,26)(21,27)(22,24)(25,32)(28,29)(30,31);(1,2)(3,4)(17,20,22,23)(18,24,26,19)(21,28,31,32)(25,30,29,27);(9,11)(10,14)(12,15)(13,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,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, 4220102);