#5020748

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

4065 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^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	#72304047, #72304086, #72304088, …
maximal normal subgroup of	#72304047, #72304086, #72304088, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4927944);