#950631

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2808 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)(3,5)(4,7)(6,8)(9,10,13,14)(11,12,15,16)(18,21)(22,24)(26,30)(28,31);(2,3)(6,7)(10,14)(11,15)(17,18)(19,22)(20,21)(23,24)(25,26,29,30)(27,31,32,28);(2,3)(6,7)(10,14)(11,15)(19,23)(22,24)(26,30)(27,32);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24)(25,28,29,31)(26,27,30,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303616, #72303651, #72303653
maximal normal subgroup of	#72303616, #72303651, #72303653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 857827);