#3907232

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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2540 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)(9,10,13,12)(11,14,15,16)(17,18,22,21)(19,20,24,23)(25,26)(27,32)(28,31)(29,30);(2,5)(3,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,24)(18,23,21,20)(26,30)(31,32);(3,8)(6,7)(10,12)(14,16)(18,21)(19,24)(25,27,29,28)(26,31,30,32);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23)(25,28,29,27)(26,32,30,31);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,14,15,16)(17,21,22,18)(19,23,24,20)(25,26)(27,32)(28,31)(29,30)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=gh,d^2=gh,e^2=fgh,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	#72232690, #72232691, #72232694
maximal normal subgroup of	#72232690, #72232691, #72232694

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3814428);