#397908

order	512
nilpotency class	3
isoclinism	isoclinic class of #394378
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 305104);