#1297990

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2716 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fh,d^2=i,e^2=h,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, 1205186);