#727128

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3702 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(18,21)(22,23)(25,26)(27,32)(28,30)(29,31);(2,5)(6,8)(10,14)(12,16)(17,18)(19,23)(20,21)(22,24)(25,27,29,28)(26,30,31,32);(3,7)(6,8)(10,14)(11,15)(18,21)(22,23)(27,28)(30,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,15)(10,12,14,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,27)(26,30,31,32);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,14)(13,16)(17,19)(18,22)(20,24)(21,23)(25,26)(27,32)(28,30)(29,31)

Polycyclic presentation:

2^9|b^2=ghi,d^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329098, #72329151, #72329171
maximal normal subgroup of	#72329098, #72329151, #72329171
maximal quotient of	#72330300

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 634324);