#1071550

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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	#16258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2614 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=f,d^2=f,e^2=ghi,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, 978746);