#4117637

order	512
nilpotency class	2
isoclinism	isoclinic class of #4033837
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2586 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=fhi,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303941, #72304135, #72304136
maximal normal subgroup of	#72303941, #72304135, #72304136

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4024833);