#385805

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

979 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)(8,20)(9,12)(10,26)(11,16)(13,23)(14,19)(15,24)(18,31)(21,30)(22,32)(25,29)(27,33)(28,35)(34,36);(5,7)(6,12)(8,18,23,36)(9,24)(10,25)(11,19)(13,30,20,28)(14,15)(16,29)(17,26)(21,27,35,22)(31,33,34,32);(5,8,10,27,11,23,24,22)(6,13,15,33,16,20,26,32)(7,18,9,21,19,36,25,35)(12,28,17,34,29,30,14,31);(5,9,24,7,11,25,10,19)(6,14,26,29,16,17,15,12)(8,21,22,18,23,35,27,36)(13,31,32,30,20,34,33,28)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ef,d^2=efi,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 293001);