#1084734

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#534	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(2,4)(6,8)(13,14,17,20)(15,16,19,18);(10,12)(13,15,17,19)(14,18,20,16);(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,16,17,18)(14,19,20,15);(5,6)(7,8)(13,16)(14,15)(17,18)(19,20);(1,2)(3,4)(5,6)(7,8)(13,15)(14,16)(17,19)(18,20)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 991930);