#391674

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

681 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)(5,6,10,12,11,16,9,15)(7,17,21,29,18,28,20,13)(8,14,24,26,22,32,25,31)(19,30,23,35,33,36,34,27);(6,12)(7,18)(8,22)(9,10)(13,17)(14,26)(15,16)(19,33)(27,36)(28,29)(30,35)(31,32);(5,7,9,20,11,18,10,21)(6,13,15,28,16,29,12,17)(8,23,24,33,22,34,25,19)(14,30,26,35,32,36,31,27);(5,8,10,24,11,22,9,25)(6,14,12,26,16,32,15,31)(7,19,21,23,18,33,20,34)(13,27,17,30,29,35,28,36)

Polycyclic presentation:

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

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, 298870);