#397208

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
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	#60423	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1554 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,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, 304404);