#669474

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2381 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fgi,d^2=fgh,e^2=fgh,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, 576670);