#364715

order	512
nilpotency class	3
isoclinism	isoclinic class of #23918
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271911);