#422762

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

1976 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)(7,16)(8,13)(9,26)(10,14)(11,15)(12,19)(17,20)(18,33)(21,28)(22,36)(23,32)(24,30)(25,31)(27,35)(29,34);(5,7,9,19,11,20,10,21)(6,12,14,16,15,28,26,17)(8,22,23,34,25,35,24,18)(13,29,30,36,31,33,32,27);(5,6)(7,17,20,16)(8,13)(9,26)(10,14)(11,15)(12,19,28,21)(18,29,34,33)(22,27,35,36)(23,32)(24,30)(25,31);(2,4)(5,8,11,25)(6,13,15,31)(7,18,20,34)(9,24,10,23)(12,27,28,36)(14,32,26,30)(16,33,17,29)(19,35,21,22)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313084, #72313145, #72313166
maximal normal subgroup of	#72313084, #72313145, #72313166

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329958);