#148302

order	512
nilpotency class	4
isoclinism	isoclinic class of #13396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

676 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)(5,6)(7,16,19,31)(8,22)(9,14)(10,27)(11,15)(12,18,28,36)(13,25)(17,33,29,30)(20,32,21,35)(23,24)(26,34);(5,7,8,18,10,21,25,33,11,19,26,36,9,20,24,30)(6,12,13,16,14,29,22,32,15,28,23,31,27,17,34,35);(2,4)(5,6)(7,17)(8,23)(9,27)(10,14)(11,15)(12,21)(13,26)(16,30)(18,32)(19,29)(20,28)(22,24)(25,34)(31,33)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249883, #72249891, #72249947, …
maximal normal subgroup of	#72249883, #72249891, #72249947, …
maximal quotient of	#72264800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55498);