#420428

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

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

1676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 327624);