#461970

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369166);