#476176

order	512
nilpotency class	3
isoclinism	isoclinic class of #476175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1313 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)(3,4)(5,6,9,15,11,17,10,16)(7,18,20,13,22,29,21,12)(8,23,25,33,27,14,26,32)(19,31,24,28,35,36,34,30);(1,2)(3,4)(5,7,11,22)(6,12,17,13)(8,19,27,35)(9,21,10,20)(14,28,23,30)(15,29,16,18)(24,25,34,26)(31,33,36,32);(2,4)(5,7)(6,13)(8,24)(9,21)(10,20)(11,22)(12,17)(14,31)(15,18)(16,29)(19,25)(23,36)(26,35)(27,34)(28,33)(30,32);(2,4)(5,8)(6,14)(7,19)(9,26)(10,25)(11,27)(12,28)(13,30)(15,33)(16,32)(17,23)(18,31)(20,34)(21,24)(22,35)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383372);