#678224

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

1938 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)(3,8)(4,7)(5,6)(9,10)(11,13)(12,15)(14,16)(17,18,21,20)(19,23,24,22)(25,26,30,29)(27,32,28,31);(2,6)(7,8)(9,11,14,12)(10,15,16,13)(18,20)(22,23)(25,27,30,28)(26,31,29,32);(1,3,5,4)(2,7,6,8)(9,11,14,12)(10,13,16,15)(17,18,21,20)(19,22,24,23)(25,26,30,29)(27,32,28,31);(1,4,5,3)(2,8,6,7)(9,12,14,11)(10,15,16,13)(17,19)(18,22)(20,23)(21,24)(25,28,30,27)(26,31,29,32);(1,2)(3,7)(4,8)(5,6)(9,13,14,15)(10,11,16,12)(17,20,21,18)(19,23,24,22)(25,29,30,26)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=fgi,e^2=fg,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, 585420);