#615593

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

2434 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)(3,7)(9,10,14,12)(11,16,13,15)(18,20)(22,23)(26,30)(27,28);(1,2,5,4)(3,6,7,8)(9,11,14,13)(10,15,12,16)(17,18,21,20)(19,22,24,23)(25,26,29,30)(27,31,28,32);(9,11,14,13)(10,15,12,16)(17,19,21,24)(18,22,20,23)(25,26)(27,31)(28,32)(29,30);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,16,13,15)(25,27,29,28)(26,31,30,32);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,16,12,15)(17,20,21,18)(19,23,24,22)(25,28,29,27)(26,32,30,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522789);