#2920870

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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

2360 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=gh,c^2=ghi,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=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, 2828066);