#517610

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

2750 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=g,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, 424806);