#2542316

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=fhi,d^2=f,e^2=hi,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.

Relationships

maximal subgroup of	#72334167, #72334187, #72334213
maximal normal subgroup of	#72334167, #72334187, #72334213

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2449512);