#1494100

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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

1802 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fh,c^2=fghi,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef

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, 1401296);