#419970

order	512
nilpotency class	3
isoclinism	isoclinic class of #417696
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327166);