#391220

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1082 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)(7,19,23,13)(8,24)(9,15)(10,18)(11,17)(12,16)(14,26)(20,33,36,30)(21,31,22,32)(25,29,28,35)(27,34);(1,2)(3,4)(5,7)(6,13)(8,25)(9,28)(10,22)(11,21)(12,23)(14,33)(15,29)(16,19)(17,32)(18,31)(20,27)(24,35)(26,36)(30,34);(2,4)(5,8,12,9)(6,14,16,34)(7,20,23,36)(10,27,11,26)(13,29,19,35)(15,18,24,17)(21,25,22,28)(30,31,33,32);(5,9,12,8)(6,15,16,24)(7,20,23,36)(10,26,11,27)(13,30,19,33)(14,17,34,18)(21,25,22,28)(29,32,35,31)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,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, 298416);