#381485

order	512
nilpotency class	3
isoclinism	isoclinic class of #381483
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14299
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288681);