#351119

order	512
nilpotency class	3
isoclinism	isoclinic class of #347758
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

605 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,g^d=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258315);