#148219

order	512
nilpotency class	4
isoclinism	isoclinic class of #13442
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
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

682 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)(5,6,10,14,9,15,24,30,12,16,26,25,11,8,22,23)(7,17,19,32,18,13,27,28,21,31,34,36,20,29,33,35);(5,7,9,18,12,21,11,20)(6,13,8,17,16,29,15,31)(10,19,24,27,26,34,22,33)(14,28,23,32,25,35,30,36);(2,4)(5,8)(6,11)(7,13)(9,16)(10,25)(12,15)(14,26)(17,18)(19,32)(20,31)(21,29)(22,23)(24,30)(27,35)(28,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321498, #72321547, #72321592, …
maximal normal subgroup of	#72321498, #72321547, #72321592, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55415);