#829482

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2205 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)(3,7)(4,6)(5,8)(9,10,13,12)(11,15,16,14)(17,18,22,21)(19,24,20,23)(25,26)(27,29)(28,32)(30,31);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,24)(20,23)(21,22)(25,27,30,32)(26,28,31,29);(1,3,5,4)(2,7,8,6)(9,10,13,12)(11,15,16,14)(17,19,22,20)(18,23,21,24)(25,28,30,29)(26,27,31,32);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,20,22,19)(18,23,21,24)(25,26)(27,28)(29,32)(30,31);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,21,22,18)(19,24,20,23)(25,29,30,28)(26,27,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=f,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 736678);