#2921783

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	C₂×#2703	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3373 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)(11,15)(12,13)(14,16)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,30,32,31);(6,8)(10,13)(14,15)(18,22)(20,29)(23,25)(26,32);(5,6)(7,8)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31);(1,2,3,4)(9,11,12,16)(10,14,13,15)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32);(9,10)(11,15)(12,13)(14,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gi,e^2=g,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, 2828979);