#4942366

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3632 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)(9,10,13,12)(11,15,16,14)(17,18)(19,26)(20,23)(21,29)(22,25)(24,30)(27,31)(28,32);(10,12)(14,15)(17,19,22,20)(18,23,25,26)(21,28,30,27)(24,32,29,31);(2,3)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30);(1,3,4,2)(5,6)(7,8)(9,11,13,16)(10,14,12,15)(17,20,22,19)(18,23,25,26)(21,28,30,27)(24,31,29,32);(9,12,13,10)(11,15,16,14)(17,21,22,30)(18,24,25,29)(19,27,20,28)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=h,d^2=fg,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, 4849562);