#494800

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=e,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	#72265161, #72265169, #72265201, …
maximal normal subgroup of	#72265161, #72265169, #72265201, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401996);