#829494

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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2233 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)(4,8)(5,6)(10,11)(14,15)(17,18,20,23)(19,22,24,21)(25,26,29,28)(27,30,32,31);(2,6)(4,7)(9,10,13,11)(12,15,16,14)(17,19,20,24)(18,21,23,22)(25,26)(27,31)(28,29)(30,32);(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32);(1,3)(2,7)(4,6)(5,8)(9,11,13,10)(12,14,16,15)(17,18)(19,21)(20,23)(22,24)(25,27)(26,31)(28,30)(29,32);(1,4,5,7)(2,8,6,3)(9,12,13,16)(10,14,11,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=hi,e^2=fg,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, 736690);