#511883

order	512
nilpotency class	4
isoclinism	isoclinic class of #511751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6)(7,15)(8,11)(9,21)(10,24)(12,17)(13,23)(14,20)(16,31)(18,33)(19,34)(22,30)(25,29)(26,32)(27,36)(28,35);(1,2,3,4)(6,11)(12,26)(13,27)(14,24)(15,31)(21,34)(28,30)(29,33);(5,7)(6,12)(8,16)(9,22)(10,25)(11,26)(13,29)(14,30)(15,21)(17,23)(18,20)(19,35)(24,28)(27,33)(31,34)(32,36);(9,19)(13,27)(15,31)(17,32)(21,34)(22,35)(23,36)(29,33)

Polycyclic presentation:

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

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, 419079);