#494801

order	512
nilpotency class	4
isoclinism	isoclinic class of #33638
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250713, #72250721, #72250729, …
maximal normal subgroup of	#72250713, #72250721, #72250729, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401997);