#500682

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1245 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265609, #72265615, #72265636
maximal normal subgroup of	#72265609, #72265615, #72265636

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407878);