#854972

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2188 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=h,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 762168);