#372738

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1451 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279934);