#391633

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

557 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:

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

Polycyclic presentation:

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

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, 298829);