#864885

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂³×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15123	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2464 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 772081);