#5022312

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2831 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262779, #72262783, #72262799, …
maximal normal subgroup of	#72262779, #72262783, #72262799, …
maximal quotient of	#72338910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4929508);