#4929616

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222749
maximal normal subgroup of	#72222749
maximal quotient of	#72222692

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836812);