#2052328

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1697 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=fghi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1959524);