#148079

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	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

680 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=i,e^2=hi,g^2=i,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, #72224936, #72226807, …
maximal normal subgroup of	#72224839, #72224936, #72226807, …
maximal quotient of	#72224834

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55275);