#557071

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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

2518 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 464267);