#1499561

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=gi,d^2=fghi,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, 1406757);