#157455

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

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

2832 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,d^2=f,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, 64651);