#4933772

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
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	C₂×#2430	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3406 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)(9,10,13,15)(11,12,16,14)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(9,11)(10,12)(13,16)(14,15)(18,19)(21,29)(23,25)(27,32);(6,8)(9,11,13,16)(10,14,15,12)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31);(1,2)(3,4)(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(9,12)(10,11)(13,14)(15,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^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, 4840968);