#2921330

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2326 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)(9,10,12,14)(11,13,16,15)(17,18)(19,24)(20,28)(21,31)(22,32)(23,25)(26,27)(29,30);(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,25,28)(21,29,32,26)(22,27,31,30);(5,6)(7,8)(17,20,23,19)(18,24,25,28)(21,29,32,26)(22,30,31,27);(1,2,3,4)(9,10)(11,15)(12,14)(13,16)(17,21)(18,22)(19,26)(20,29)(23,32)(24,30)(25,31)(27,28);(9,11)(10,13)(12,16)(14,15)(17,22)(18,21)(19,27)(20,30)(23,31)(24,29)(25,32)(26,28)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=fgi,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, 2828526);