#2826902

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2492 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,d^2=fgh,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, 2734098);