#1494092

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

2070 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)(3,4,8,7)(10,11)(12,16)(17,18)(19,22)(20,21)(23,24)(25,26,29,32)(27,28,31,30);(1,3,6,8)(2,7,5,4)(9,10,13,11)(12,15,16,14)(18,20)(22,24)(25,27,29,31)(26,30,32,28);(9,10,13,11)(12,14,16,15)(17,18,21,20)(19,22,23,24)(25,26)(27,30)(28,31)(29,32);(1,4)(2,3)(5,8)(6,7)(9,11,13,10)(12,14,16,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,30)(26,31,32,27);(1,5,6,2)(3,7,8,4)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,22,23,24)(25,26)(27,28)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fh,c^2=fghi,d^2=gh,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, 1401288);