#2760745

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2401 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)(9,10,13,16)(11,15,12,14)(17,18)(19,23)(20,22)(21,28)(24,29)(25,31)(26,27)(30,32);(6,8)(9,10,13,16)(11,14,12,15)(18,20)(23,27)(24,28)(30,31);(5,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19,22,26)(18,23,20,27)(21,25,29,32)(24,30,28,31);(1,2)(3,4)(9,10)(11,14)(12,15)(13,16)(17,20,22,18)(19,23,26,27)(21,24,29,28)(25,31,32,30);(9,12,13,11)(10,15,16,14)(17,21,22,29)(18,24,20,28)(19,25,26,32)(23,30,27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fg,d^2=f,e^2=fg,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, 2667941);