#2415741

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

2108 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,16,12,14)(17,18)(19,23)(20,24)(21,22)(25,26)(27,28)(29,32)(30,31);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,29)(28,31)(30,32);(3,7)(6,8)(11,12)(14,16)(18,22)(19,24)(25,28,30,29)(26,27,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,20)(18,19)(21,23)(22,24)(25,29,30,28)(26,27,31,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,14,15,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=hi,e^2=gh,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, 2322937);