#4943997

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3636 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)(3,4)(5,6,7,8)(9,10)(11,15)(12,14)(13,16)(18,20)(23,25)(24,28)(30,31);(9,10)(11,13)(12,14)(15,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31);(2,3)(9,10,12,14)(11,13,16,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,3,4,2)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31);(9,11)(10,13)(12,16)(14,15)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=h,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4851193);