#419998

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
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	#60425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1398 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)(5,6,11,28,9,15,10,16)(7,17,22,31,20,12,21,32)(8,23,27,13,25,33,26,14)(18,30,24,35,36,34,19,29);(2,3)(5,7)(6,12)(8,24)(9,20)(10,21)(11,22)(13,30)(14,34)(15,17)(16,31)(18,26)(19,25)(23,29)(27,36)(28,32)(33,35);(5,8)(6,13)(7,18)(9,25)(10,27)(11,26)(12,29)(14,15)(16,33)(17,35)(19,22)(20,36)(21,24)(23,28)(30,31)(32,34);(1,3,4,2)(5,8)(6,14)(7,19)(9,25)(10,26)(11,27)(12,30)(13,15)(16,33)(17,34)(18,22)(20,24)(21,36)(23,28)(29,31)(32,35)

Polycyclic presentation:

2^9|a^2=ef,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,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, 327194);