#392704

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3713 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)(5,6,10,12,11,16,9,15)(7,17,20,13,21,27,19,28)(8,22,24,14,25,29,23,30)(18,32,34,36,35,26,33,31);(1,2,3,4)(6,12)(9,10)(13,17)(14,22)(15,16)(19,20)(23,24)(26,31)(27,28)(29,30)(32,36)(33,34);(5,7)(6,13)(8,18)(9,20)(10,19)(11,21)(12,17)(14,26)(15,27)(16,28)(22,31)(23,34)(24,33)(25,35)(29,36)(30,32);(5,8)(6,14)(7,18)(9,24)(10,23)(11,25)(12,22)(13,26)(15,29)(16,30)(17,31)(19,34)(20,33)(21,35)(27,36)(28,32)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299900);