#459300

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1159 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:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 366496);