#2142122

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14885	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2708 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)(3,7)(10,11)(12,16)(17,18,23,24)(19,21,22,20)(25,26,28,30)(27,31,32,29);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(17,19,23,22)(18,20,24,21)(25,26,28,30)(27,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,10,13,11)(12,15,16,14)(17,20,23,21)(18,19,24,22)(25,27)(26,29)(28,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,21,23,20)(18,19,24,22)(25,26)(27,29)(28,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,16)(10,14,11,15)(17,22,23,19)(18,20,24,21)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,c^2=f,d^2=f,e^2=f,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, 2049318);