#158163

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

2404 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=efgh,d^2=eg,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, 65359);