#422950

order	512
nilpotency class	3
isoclinism	isoclinic class of #420453
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

2002 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)(5,6,9,12)(7,17,20,27)(8,14,21,26)(10,16,11,15)(13,23,25,18)(19,31,34,36)(22,30,24,29)(28,33,35,32);(6,12)(7,18,20,23)(8,21)(13,27,25,17)(15,16)(19,32,34,33)(22,24)(28,36,35,31);(5,7,10,18,9,20,11,23)(6,13,15,17,12,25,16,27)(8,19,22,32,21,34,24,33)(14,28,29,31,26,35,30,36);(2,4)(5,8)(6,14)(7,19)(9,21)(10,22)(11,24)(12,26)(13,28)(15,29)(16,30)(17,31)(18,32)(20,34)(23,33)(25,35)(27,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287064, #72287334, #72287352
maximal normal subgroup of	#72287064, #72287334, #72287352

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330146);