#331019

order	512
nilpotency class	3
isoclinism	isoclinic class of #331007
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

484 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)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,21,25,23,26,22,20)(19,27,28,32,30,31,29,24);(1,3,4,10)(2,6,7,14)(5,9,12,11)(8,16,15,13)(17,19,23,30)(18,24,26,32)(20,27,25,31)(21,29,22,28);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(18,20)(19,28)(21,22)(25,26)(27,31)(29,30);(17,20,22,26,23,25,21,18)(19,27,28,32,30,31,29,24)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238215);