#4320034

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

2721 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26)(27,30)(28,32)(29,31);(1,3,5,8)(2,6,7,4)(9,11,13,16)(10,12,14,15)(18,21)(22,24)(25,26,29,31)(27,28,32,30);(2,7)(4,6)(10,14)(11,16)(17,19,20,23)(18,22,21,24)(25,27,29,32)(26,30,31,28);(1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,28)(29,32)(30,31);(1,3)(2,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,28)(26,27)(29,30)(31,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326557, #72326584, #72326609
maximal normal subgroup of	#72326557, #72326584, #72326609
maximal quotient of	#72328394

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4227230);