#148570

order	512
nilpotency class	4
isoclinism	isoclinic class of #13373
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#486
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#2041
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 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)(7,20,15,29)(8,26,30,23)(9,32)(10,21)(11,17)(12,27)(13,19)(14,22,31,24)(16,34)(18,33)(25,35,36,28);(5,7)(6,14)(8,27)(9,26)(10,22)(11,23)(12,36)(13,15)(16,35)(17,20)(18,30)(19,31)(21,28)(24,34)(25,33)(29,32);(2,4)(5,8,9,28,12,31,34,29,13,30,11,35,33,14,10,20)(6,15,16,24,18,36,32,26,19,7,21,22,27,25,17,23)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249923, #72249938, #72249966, …
maximal normal subgroup of	#72249923, #72249938, #72249966, …
maximal quotient of	#72308751

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55766);