#1033751

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

1904 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=g,d^2=f,e^2=h,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, 940947);