#2542919

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1958 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=ghi,d^2=f,e^2=ghi,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, 2450115);