#456518

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

1126 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)(7,17)(8,22,25,29)(9,15)(10,13)(11,16)(12,20)(14,24,30,23)(18,27)(19,33,32,26)(21,28)(31,35,36,34);(1,2)(3,4)(5,7,9,20,11,18,10,21)(6,12,13,17,16,28,15,27)(8,19,24,34,25,32,23,35)(14,26,29,31,30,33,22,36);(2,4)(6,13)(7,18)(8,23)(9,10)(12,17)(15,16)(19,34)(22,29)(24,25)(27,28)(31,36)(32,35);(5,8)(6,14)(7,19)(9,24)(10,23)(11,25)(12,26)(13,29)(15,22)(16,30)(17,31)(18,32)(20,34)(21,35)(27,36)(28,33)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,e^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363714);