#411840

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2024 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319036);