#495846

order	512
nilpotency class	4
isoclinism	isoclinic class of #483882
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

674 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320110, #72320189, #72320324, …
maximal normal subgroup of	#72320110, #72320189, #72320324, …
maximal quotient of	#72343994

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403042);