#2804845

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

2024 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=fh,d^2=h,e^2=fghi,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, 2712041);