#147006

order	512
nilpotency class	4
isoclinism	isoclinic class of #13291
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1993
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

554 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224893, #72224939, #72226807, …
maximal normal subgroup of	#72224893, #72224939, #72226807, …
maximal quotient of	#72224889, #72240742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54202);