#304369

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

931 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211565);