#759048

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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	#15273	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2741 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=hi,e^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	#72263189, #72263204, #72263208
maximal normal subgroup of	#72263189, #72263204, #72263208

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 666244);