#450133

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1009 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:

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

Polycyclic presentation:

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

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, 357329);