#436468

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1178 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ef,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 343664);