#391600

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₄³

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

807 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)(5,6,10,15,9,16,11,13)(7,12,17,26,19,27,20,32)(8,14,23,30,24,31,22,28)(18,25,33,29,34,36,21,35);(5,7,9,19)(6,12,16,27)(8,21,24,33)(10,20,11,17)(13,26,15,32)(14,29,31,35)(18,23,34,22)(25,28,36,30);(6,13)(7,17)(10,11)(14,30)(15,16)(18,33)(19,20)(21,34)(22,23)(25,36)(26,32)(28,31);(5,8,10,23,9,24,11,22)(6,14,15,30,16,31,13,28)(7,18,17,33,19,34,20,21)(12,25,26,29,27,36,32,35)

Polycyclic presentation:

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

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, 298796);