#147195

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

518 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)(7,17,20,34)(8,23)(9,13)(10,27)(11,16)(12,19,29,36)(14,24)(15,25)(18,21,31,33)(22,30,28,35)(26,32);(5,7,9,21,25,28,8,19,11,20,26,33,10,22,24,36)(6,12,14,30,27,18,32,34,16,29,23,35,15,31,13,17);(2,4)(5,6)(7,18,20,31)(8,13)(9,23)(10,15)(11,16)(12,28,29,22)(14,26)(17,33,34,21)(19,30,36,35)(24,32)(25,27)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,g^2=hi,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	#72250035, #72250040, #72250077, …
maximal normal subgroup of	#72250035, #72250040, #72250077, …
maximal quotient of	#72264886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54391);