#148077

order	512
nilpotency class	4
isoclinism	isoclinic class of #134415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

736 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224839, #72224939, #72226810, …
maximal normal subgroup of	#72224839, #72224939, #72226810, …
maximal quotient of	#72224835

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55273);