#363247

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=gi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270443);