#940595

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
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	#15124	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2692 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)(10,12)(14,16)(17,18,21,24)(19,23,20,22)(26,28)(27,31);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,15,16)(17,19,21,20)(18,22,24,23)(25,26,29,28)(27,32,31,30);(2,3)(6,7)(11,15)(14,16)(17,18,21,24)(19,22,20,23)(25,26,29,28)(27,32,31,30);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,27,29,31)(26,30,28,32);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,20,21,19)(18,23,24,22)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=hi,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 847791);