#385532

order	512
nilpotency class	3
isoclinism	isoclinic class of #30437
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1147 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)(7,17)(8,14)(9,24)(10,27)(11,16)(12,20)(13,23)(15,28)(18,32)(19,34)(21,31)(22,30)(25,29)(26,33)(35,36);(5,7)(6,12)(8,18)(9,25)(10,21)(11,22)(13,32)(14,34)(15,30)(16,31)(17,27)(19,26)(20,28)(23,36)(24,35)(29,33);(5,8,11,9)(6,13,16,33)(7,18)(10,26,28,23)(12,29)(14,15,24,27)(17,35)(19,21)(20,36)(22,25)(30,34)(31,32);(5,9,11,8)(6,14,16,24)(7,19)(10,26,28,23)(12,29)(13,15,33,27)(17,34)(18,21)(20,25)(22,36)(30,35)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292728);