#148114

order	512
nilpotency class	4
isoclinism	isoclinic class of #13374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
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

706 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)(5,6)(7,16,18,35)(8,21)(9,25)(10,28)(11,15)(12,19,30,17)(13,22)(14,24)(20,32,29,36)(23,34,27,31)(26,33);(1,2,3,4)(7,17,18,19)(8,22)(9,26)(10,24)(12,16,30,35)(13,21)(14,28)(20,27,29,23)(25,33)(31,32,34,36);(5,7)(6,12)(8,23)(9,27)(10,29)(11,18)(13,32)(14,34)(15,30)(16,25)(17,22)(19,26)(20,24)(21,35)(28,31)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249891, #72249898, #72249931, …
maximal normal subgroup of	#72249891, #72249898, #72249931, …
maximal quotient of	#72264826, #72307612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55310);