#1168418

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2763 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)(9,10,14,12)(11,16,15,13)(17,18,22,23)(19,20,26,24)(21,29,30,25)(27,31,32,28);(6,8)(18,23)(19,26)(21,30)(28,31);(1,2,4,3)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,26)(18,24,23,20)(21,27,30,32)(25,31,29,28);(1,2,4,3)(9,12,14,10)(11,16,15,13)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32);(1,3,4,2)(9,13,14,16)(10,15,12,11)(17,21,22,30)(18,25,23,29)(19,27,26,32)(20,28,24,31)

Polycyclic presentation:

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

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, 1075614);