#615595

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

2234 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=i,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, 522791);