#158392

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=eg,d^2=eh,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, 65588);