#391992

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

717 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=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, 299188);