#624978

order	512
nilpotency class	2
isoclinism	isoclinic class of #616960
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2052 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)(9,10,13,14)(11,12,15,16)(18,20)(19,23)(26,29)(30,32);(2,6)(3,7)(10,14)(11,15)(17,18,21,20)(19,24,23,22);(9,11)(10,12)(13,15)(14,16)(17,18,21,20)(19,22,23,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,7)(2,4,6,8)(9,11)(10,12)(13,15)(14,16)(17,19,21,23)(18,22,20,24);(1,4)(2,7)(3,6)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20,21,18)(19,24,23,22)(25,27,28,31)(26,30,29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df

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, 532174);