#391578

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

683 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,f^c=fi,e^d=ei,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, 298774);