#2109054

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2456	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2016250);