#2306468

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2404 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)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,21)(22,23)(25,29)(26,31)(27,30)(28,32);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(19,26)(20,29)(28,30);(5,6)(7,8)(10,13)(14,15)(17,19,22,26)(18,24,23,31)(20,27,29,32)(21,30,25,28);(1,2,3,4)(9,11)(10,14)(12,16)(13,15)(17,20)(18,25)(19,27)(21,23)(22,29)(24,28)(26,32)(30,31);(9,10)(11,14)(12,13)(15,16)(17,21,22,25)(18,20,23,29)(19,28,26,30)(24,32,31,27)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=fg,d^2=fi,e^2=fg,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, 2213664);