#464959

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1139 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:

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

Polycyclic presentation:

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

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, 372155);