#899382

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2312 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)(11,13)(12,15)(14,16)(17,18,22,21)(19,24,23,20)(25,26,29,28)(27,30,31,32);(1,2,5,4)(3,6,7,8)(10,11)(14,15)(19,23)(20,24)(27,31)(30,32);(9,10,13,11)(12,14,16,15)(17,19)(18,20)(21,24)(22,23)(25,27)(26,30)(28,32)(29,31);(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,15,16,14)(17,20,22,24)(18,23,21,19);(1,4,5,2)(3,8,7,6)(9,12)(10,14)(11,15)(13,16)(17,21,22,18)(19,24,23,20)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=fh,b^2=gi,c^2=f,d^2=g,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, 806578);