#1294820

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

2886 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)(6,7)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,29)(21,26)(22,23)(25,30)(27,31)(28,32);(9,10,12,14)(11,13,16,15)(18,23)(19,27)(25,29)(26,28);(1,2,4,3)(5,6,8,7)(9,10)(11,13)(12,14)(15,16)(17,19,22,27)(18,24,23,31)(20,28,30,26)(21,29,32,25);(5,6,8,7)(17,20,22,30)(18,25,23,29)(19,28,27,26)(21,31,32,24);(1,3,4,2)(5,7,8,6)(9,11,12,16)(10,13,14,15)(17,21)(18,26)(19,29)(20,24)(22,32)(23,28)(25,27)(30,31)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=gh,d^2=i,e^2=g,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, 1202016);