#3966460

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2882 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=f,d^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326557, #72326602, #72326622
maximal normal subgroup of	#72326557, #72326602, #72326622
maximal quotient of	#72328811

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3873656);