#899993

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2339 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)(11,15)(12,14)(13,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,31)(30,32);(1,2,5,4)(3,6,7,8)(9,10,12,14)(11,13,16,15)(18,22)(19,24)(25,27,29,32)(26,28,31,30);(9,11)(10,13)(12,16)(14,15)(17,19,21,24)(18,23,22,20)(25,26,29,31)(27,30,32,28);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,31)(29,32);(1,4,5,2)(3,8,7,6)(9,11)(10,13)(12,16)(14,15)(17,20,21,23)(18,24,22,19)(25,28,29,30)(26,27,31,32)

Polycyclic presentation:

2^9|b^2=ghi,c^2=h,e^2=fghi,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, 807189);