#3547078

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3234 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)(9,10,13,16)(11,12,15,14)(17,18)(19,24)(20,21)(22,23)(25,26,28,30)(27,31,32,29);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,16)(13,15)(18,23)(19,20)(27,32)(29,31);(2,6)(5,7)(11,15)(12,14)(17,19,22,20)(18,24,23,21)(25,26)(27,29)(28,30)(31,32);(1,3)(2,7)(4,8)(5,6)(9,10)(11,12)(13,16)(14,15)(17,20,22,19)(18,21,23,24)(25,27)(26,29)(28,32)(30,31);(1,2)(3,5)(4,6)(7,8)(9,12)(10,15)(11,16)(13,14)(17,21)(18,20)(19,23)(22,24)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334184, #72334201, #72334234
maximal normal subgroup of	#72334184, #72334201, #72334234

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3454274);