#557026

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	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,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, 464222);