#2760388

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3309 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)(10,13)(11,15)(17,18,23,22)(19,25,20,24)(21,29,30,26)(27,31,28,32);(6,8)(10,13)(14,16)(17,19,23,20)(18,24,22,25)(21,27,30,28)(26,31,29,32);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32);(9,11)(10,14)(12,15)(13,16)(17,22,23,18)(19,25,20,24)(21,26,30,29)(27,31,28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2667584);