#4929060

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2812 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:

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

Polycyclic presentation:

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

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, 4836256);