#471453

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1113 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)(6,12)(7,17)(8,21)(9,10)(15,16)(18,28)(20,25)(22,33)(23,24)(26,31)(27,36)(30,35);(5,6,9,12,11,16,10,15)(7,18,25,29,20,28,17,13)(8,14,21,26,24,32,23,31)(19,30,22,35,34,36,33,27);(2,4)(5,7)(6,13)(8,22)(9,25)(10,17)(11,20)(12,18)(14,30)(15,28)(16,29)(19,23)(21,34)(24,33)(26,35)(27,31)(32,36);(5,8)(6,14)(7,19)(9,21)(10,23)(11,24)(12,26)(13,27)(15,31)(16,32)(17,33)(18,30)(20,34)(22,25)(28,36)(29,35)

Polycyclic presentation:

2^9|b^2=egh,e^2=i,f^2=i,g^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,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, 378649);