#4930665

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 8388608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4616 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)(11,15)(12,13)(14,16)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32);(6,8)(9,11,14,13)(10,15,16,12)(18,22)(19,26)(24,28)(27,32);(9,12)(10,11)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32);(1,2,3,4)(5,6)(7,8)(9,13,14,11)(10,15,16,12)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32);(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4837861);