#2844161

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2188 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,16,13,15)(19,23)(21,24)(25,26,30,29)(27,28,32,31);(2,3)(4,8)(9,11)(10,15)(12,16)(13,14)(17,18,20,22)(19,21,23,24)(25,27,30,32)(26,31,29,28);(1,2,5,3)(4,6,8,7)(9,10,14,12)(11,16,13,15)(17,19)(18,21)(20,23)(22,24)(25,26,30,29)(27,31,32,28);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,16,13,15)(17,18)(19,21)(20,22)(23,24)(25,28)(26,27)(29,32)(30,31);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,15,12,16)(17,19)(18,21)(20,23)(22,24)(25,29,30,26)(27,31,32,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=f,d^2=h,e^2=f,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, 2751357);