#1082872

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,4)(9,10,12,14)(11,15,16,13)(18,19)(20,27)(21,29)(30,31);(6,8)(11,16)(13,15)(17,18,22,19)(20,23,27,25)(21,26,29,24)(28,31,32,30);(5,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31);(1,2)(3,4)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(9,10)(11,13)(12,14)(15,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=g,e^2=f,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, 990068);