#845126

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2174 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)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,21)(20,22)(23,24)(25,26,30,32)(27,28,29,31);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,20,23)(18,21,22,24)(25,27,30,29)(26,31,32,28);(1,3,5,4)(2,6,7,8)(11,15)(13,16)(25,28,30,31)(26,27,32,29);(1,4,5,3)(2,6,7,8)(18,22)(21,24)(25,27,30,29)(26,31,32,28);(25,29,30,27)(26,31,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303962, #72303964, #72304095
maximal normal subgroup of	#72303962, #72303964, #72304095

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752322);