#158253

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=e,c^2=efgh,d^2=fg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 65449);