#862636

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1830 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=ghi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 769832);