#669019

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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

2012 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,22)(19,20,23,24)(25,26)(27,29)(28,30)(31,32);(2,6)(4,7)(10,14)(11,16)(18,22)(20,24)(26,27)(28,32);(9,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,20,23,24)(25,27,29,26)(28,31,32,30);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,15)(14,16)(17,19,21,23)(18,20,22,24);(1,4,5,7)(2,3,6,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,23)(19,22)(21,24)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=gh,c^2=fi,d^2=gh,e^2=fh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 576215);