#428073

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:

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

Polycyclic presentation:

2^9|a^2=i,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, 335269);