#148512

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=hi,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,h^b=hi,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321321, #72321352, #72321421, …
maximal normal subgroup of	#72321321, #72321352, #72321421, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55708);