#2806197

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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fg,e^2=fg,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, 2713393);