#877261

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2264 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)(3,7)(4,8)(5,6)(11,12)(15,16)(17,18,20,21)(19,23,24,22)(25,26)(27,28)(29,30)(31,32);(2,6)(3,8)(9,10,14,13)(11,16,12,15)(18,21)(22,23)(26,30)(28,31);(1,3)(2,4)(5,8)(6,7)(9,11,14,12)(10,15,13,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32);(1,3)(2,4)(5,8)(6,7)(9,12,14,11)(10,16,13,15)(17,18)(19,22)(20,21)(23,24)(25,28,29,31)(26,27,30,32);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,16,12,15)(17,19)(18,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=fg,d^2=fi,e^2=fg,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, 784457);