#2490678

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2854 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)(9,10,12,14)(11,13,15,16)(17,18,22,23)(19,24,21,20)(25,26,29,27)(28,30,32,31);(2,3)(6,7)(17,19,22,21)(18,20,23,24)(26,27)(30,31);(4,8)(6,7)(18,23)(19,21)(25,26,29,27)(28,31,32,30);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,24)(18,19,23,21)(25,27,29,26)(28,31,32,30);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,13,14,16)(17,21,22,19)(18,24,23,20)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=gh,d^2=f,e^2=fghi,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, 2397874);