#461275

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

816 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)(7,18,22,30)(8,23)(9,14)(10,15)(11,16)(12,17)(13,21,31,27)(19,34,20,28)(24,29,35,36)(25,33)(26,32);(5,7,12,22)(6,13)(8,24,26,35)(9,20,25,19)(10,21,11,27)(14,29)(15,18)(16,30)(17,31)(23,34)(28,32)(33,36);(5,8,12,26)(6,14,17,33)(7,19,22,20)(9,10,25,11)(13,28,31,34)(15,23,16,32)(18,29,30,36)(21,35,27,24);(2,4)(5,9,10,26,12,25,11,8)(6,14,15,32,17,33,16,23)(7,20,21,35,22,19,27,24)(13,29,18,28,31,36,30,34)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368471);