#2076676

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

3545 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)(6,8)(13,14,18,21)(15,22,19,16)(17,25,26,20)(23,24,27,28);(1,2,4,3)(10,12)(15,19)(16,22)(23,27)(24,28);(2,3)(9,10)(11,12)(13,15,18,19)(14,16,21,22)(17,23,26,27)(20,24,25,28);(5,6)(7,8)(13,16)(14,19)(15,21)(17,24)(18,22)(20,27)(23,25)(26,28);(1,3,4,2)(13,17)(14,20)(15,23)(16,24)(18,26)(19,27)(21,25)(22,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^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, 1983872);