#2209479

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	#16500	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2362 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=hi,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300826, #72300857, #72300892
maximal normal subgroup of	#72300826, #72300857, #72300892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2116675);