#1499549

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

1767 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=i,d^2=fghi,e^2=fg,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, 1406745);