#1281552

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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

1838 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)(11,15)(12,13)(14,16)(17,18,21,23)(19,24,22,20)(25,26,30,28)(27,29,32,31);(9,11)(10,14)(12,15)(13,16)(17,19,21,22)(18,20,23,24)(25,26,30,28)(27,31,32,29);(1,2,5,3)(4,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,18)(19,24)(20,22)(21,23)(25,27,30,32)(26,31,28,29);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,18)(19,20)(21,23)(22,24)(25,28,30,26)(27,29,32,31);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,15,16,14)(17,20)(18,22)(19,23)(21,24)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1188748);