#455247

order	512
nilpotency class	3
isoclinism	isoclinic class of #454039
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1289 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:

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

Polycyclic presentation:

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

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, 362443);