#2831833

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2028 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,20)(19,24,21,23)(25,26,30,32)(27,29,31,28);(2,5)(3,4)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,24)(21,22)(25,27)(26,28)(29,32)(30,31);(1,3,6,4)(2,7,5,8)(9,11,13,12)(10,15,16,14)(17,18,22,20)(19,24,21,23)(25,28,30,29)(26,31,32,27);(1,4,6,3)(2,7,5,8)(9,12,13,11)(10,15,16,14)(17,20,22,18)(19,24,21,23)(25,29,30,28)(26,31,32,27);(1,5,6,2)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(17,21,22,19)(18,23,20,24)(25,26)(27,29)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=f,d^2=f,e^2=h,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, 2739029);