#3891952

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

2606 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,e^2=h,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	#72232589, #72232650, #72232691
maximal normal subgroup of	#72232589, #72232650, #72232691

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3799148);