#898729

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=f,e^2=ghi,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, 805925);