#2080866

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×#2671	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3793 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)(15,22)(16,24)(17,20)(18,21)(19,27)(23,26)(25,28);(1,2)(3,4)(10,12)(13,15,18,20)(14,17,21,22)(16,25,27,23)(19,28,24,26);(1,2,3,4)(9,10,11,12)(13,14)(15,17)(16,19)(18,21)(20,22)(23,26)(24,27)(25,28);(5,6)(7,8)(13,16,18,27)(14,19,21,24)(15,23,20,25)(17,26,22,28);(1,2)(3,4)(13,17)(14,20)(15,21)(16,26)(18,22)(19,25)(23,24)(27,28)

Polycyclic presentation:

2^9|b^2=fg,c^2=h,d^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, 1988062);