#420392

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1419 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)(5,6,10,12,11,16,9,15)(7,17,20,28,21,25,18,13)(8,14,23,26,24,30,22,29)(19,31,33,36,34,35,32,27);(6,12)(7,18)(8,22)(9,10)(15,16)(17,25)(20,21)(23,24)(26,29)(27,31)(32,33)(35,36);(5,7,9,20,11,21,10,18)(6,13,15,17,16,28,12,25)(8,19,22,33,24,34,23,32)(14,27,29,31,30,36,26,35);(2,4)(5,8,10,23,11,24,9,22)(6,14,12,26,16,30,15,29)(7,19,18,32,21,34,20,33)(13,27,25,35,28,36,17,31)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256469, #72256516, #72256521
maximal normal subgroup of	#72256469, #72256516, #72256521

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327588);