#2844148

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

1914 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgi,c^2=f,d^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, 2751344);