#428071

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1179 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,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, 335267);