#380169

order	512
nilpotency class	3
isoclinism	isoclinic class of #29736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1220 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240765, #72240768, #72240780, …
maximal normal subgroup of	#72240765, #72240768, #72240780, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287365);