#516545

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

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

4114 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239858, #72239877, #72239885
maximal normal subgroup of	#72239858, #72239877, #72239885
maximal quotient of	#72239864

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423741);