#864880

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=i,d^2=fg,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, 772076);