#2081113

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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3620 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=fgh,d^2=fg,e^2=fg,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, 1988309);