#2844107

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2029 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=f,d^2=fh,e^2=f,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, 2751303);