#554010

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

2312 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 461206);