#476145

order	512
nilpotency class	3
isoclinism	isoclinic class of #476144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1288 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)(5,6,9,14,10,12,25,15)(7,16,19,28,20,26,17,11)(8,13,22,32,24,29,23,31)(18,30,21,27,33,35,34,36);(5,7,10,20)(6,11,12,28)(8,21,24,34)(9,17,25,19)(13,30,29,35)(14,26,15,16)(18,23,33,22)(27,32,36,31);(6,12)(7,17)(8,22)(9,25)(11,26)(13,31)(16,28)(18,33)(19,20)(23,24)(27,36)(29,32);(2,4)(5,8)(6,13)(7,18)(9,23)(10,24)(11,27)(12,29)(14,31)(15,32)(16,30)(17,21)(19,34)(20,33)(22,25)(26,35)(28,36)

Polycyclic presentation:

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

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, 383341);