#148244

order	512
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

848 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312018, #72312079, #72312082, …
maximal normal subgroup of	#72312018, #72312079, #72312082, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55440);