#493828

order	512
nilpotency class	4
isoclinism	isoclinic class of #33618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

944 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)(5,6,10,14,11,8,22,15)(7,16,21,32,19,31,36,33)(9,24,17,13,25,29,34,23)(12,26,27,30,18,35,28,20);(5,7,17,26,10,21,25,30,11,19,34,35,22,36,9,20)(6,12,24,33,15,28,23,31,8,18,29,32,14,27,13,16);(2,4)(5,8)(6,11)(7,18)(9,23)(10,14)(12,19)(13,25)(15,22)(16,30)(17,29)(20,31)(21,27)(24,34)(26,32)(28,36)(33,35);(7,19)(12,18)(16,31)(20,30)(21,36)(26,35)(27,28)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321511, #72321571, #72321598, …
maximal normal subgroup of	#72321511, #72321571, #72321598, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401024);