#2306169

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2382 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)(9,10)(11,12)(13,16)(14,15)(17,18,21,22)(19,23,25,29)(20,24,28,30)(26,31,27,32);(5,6,7,8)(9,11)(10,13)(12,14)(15,16)(18,22)(19,25)(24,30)(26,27);(5,6)(7,8)(9,12,15,13)(10,16,14,11)(17,19)(18,23)(20,27)(21,25)(22,29)(24,32)(26,28)(30,31);(1,2,3,4)(9,13,15,12)(10,16,14,11)(17,18)(19,23)(20,24)(21,22)(25,29)(26,31)(27,32)(28,30);(9,14,15,10)(11,12,16,13)(17,20,21,28)(18,24,22,30)(19,26,25,27)(23,31,29,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=g,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

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, 2213365);