#1446431

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

1556 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fi,c^2=gh,d^2=fg,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, 1353627);