#1440255

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1691 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)(3,8,4,7)(9,10)(11,16)(12,14)(13,15)(17,18,21,24)(19,22,23,20)(25,26,28,30)(27,31,32,29);(1,3,6,4)(2,7,5,8)(9,11)(10,14)(12,13)(15,16)(17,19,21,23)(18,22,24,20);(1,3,6,4)(2,8,5,7)(9,10)(11,16)(12,14)(13,15)(17,20,21,22)(18,23,24,19)(25,27,28,32)(26,29,30,31);(1,4,6,3)(2,8,5,7)(9,10)(11,14)(12,16)(13,15)(25,26)(27,29)(28,30)(31,32);(1,5,6,2)(3,7,4,8)(9,12,13,11)(10,14,15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fgh,d^2=fh,e^2=f,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, 1347451);