#476522

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	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1297 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:

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

Polycyclic presentation:

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

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, 383718);