#422759

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1967 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,e^d=ei

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, 329955);