#517986

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2638 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=i,e^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=ef

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 425182);