#2807780

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2554 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)(9,10,13,15)(11,14,16,12)(17,18,22,25)(19,20,26,23)(21,29,30,24)(27,31,32,28);(6,8)(9,11,13,16)(10,12,15,14)(19,26)(20,23)(27,32)(28,31);(5,6,7,8)(9,12,13,14)(10,11,15,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,31)(28,29)(30,32);(1,2)(3,4)(9,10)(11,12)(13,15)(14,16)(17,20,22,23)(18,19,25,26)(21,28,30,31)(24,27,29,32);(9,11)(10,14)(12,15)(13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(26,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fgh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2714976);