#2608134

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

2749 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)(10,13)(14,15)(17,18)(19,27)(20,23)(21,29)(22,30)(24,28)(25,32)(26,31);(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,26,29,30);(5,6,7,8)(17,20,23,18)(19,24,28,27)(21,29,32,25)(22,30,31,26);(1,2)(3,4)(9,10)(11,15)(12,13)(14,16)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,30,27,26);(9,11)(10,14)(12,16)(13,15)(17,22)(18,26)(19,21)(20,30)(23,31)(24,29)(25,27)(28,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=h,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2515330);