#2199432

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fghi,c^2=gh,d^2=fgh,e^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2106628);