#1836041

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1974 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=i,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1743237);