#616894

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2523 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=fg,d^2=fh,e^2=fi,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, 524090);