#433830

order	512
nilpotency class	3
isoclinism	isoclinic class of #419852
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=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, 341026);