#154841

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

2334 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ef,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 62037);