#1499554

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

2174 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(19,24)(20,22)(25,26)(27,30)(28,29)(31,32);(2,5)(6,7)(10,14)(15,16)(17,18,21,23)(19,22,24,20)(26,28)(30,32);(9,11,13,12)(10,15,14,16)(17,19)(18,22)(20,23)(21,24)(25,26,29,28)(27,30,31,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,15,14,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,30)(28,32)(29,31);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,20,21,22)(18,19,23,24)(25,28,29,26)(27,30,31,32)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=ghi,e^2=g,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, 1406750);