#412037

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3060	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319233);