#1016145

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2156 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)(9,10,14,15)(11,13,12,16)(17,18,22,20)(19,24,23,21)(25,26)(27,30)(28,32)(29,31);(2,6)(3,8)(10,15)(11,12)(17,19)(18,21)(20,24)(22,23)(25,27,29,32)(26,30,31,28);(2,6)(4,7)(9,11,14,12)(10,16,15,13)(17,18,22,20)(19,24,23,21)(25,28)(26,27)(29,30)(31,32);(1,3,5,8)(2,7,6,4)(9,12,14,11)(10,16,15,13)(17,20,22,18)(19,24,23,21)(25,27)(26,28)(29,32)(30,31);(1,4,5,7)(2,8,6,3)(9,13,14,16)(10,11,15,12)(17,21,22,24)(18,23,20,19)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=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, 923341);