#439069

order	512
nilpotency class	3
isoclinism	isoclinic class of #422233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1411 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)(7,18)(8,13)(9,14)(10,16)(11,17)(12,20)(15,27)(19,33)(21,30)(22,34)(23,28)(24,35)(25,29)(26,32)(31,36);(2,4)(5,7,10,21,11,22,27,20)(6,12,16,18,17,30,15,34)(8,23,25,36,26,9,24,19)(13,31,29,14,32,33,35,28);(1,2,3,4)(5,8,11,26)(6,13,17,32)(7,19,22,36)(9,20,23,21)(10,24,27,25)(12,28,30,14)(15,29,16,35)(18,33,34,31);(5,9,10,19,11,23,27,36)(6,14,15,31,17,28,16,33)(7,8,20,24,22,26,21,25)(12,29,18,32,30,35,34,13)

Polycyclic presentation:

2^9|b^2=fg,c^2=fi,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,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, 346265);