#121968

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6,8,12)(3,9,11,13)(17,18)(19,28,32,44)(20,33)(21,26)(22,27)(23,37)(24,30,40,31)(25,36)(29,45,46,41)(34,48,47,39)(35,43)(38,42);(1,2,5,8)(3,10,11,15)(4,12,14,6)(7,13,16,9)(17,19,23,30,22,32,21,31)(18,24,37,28,27,40,26,44)(20,34,38,46,36,47,35,29)(25,41,43,39,33,45,42,48);(1,3,5,11)(2,7,8,16)(4,13,14,9)(6,15,12,10)(17,20,23,38,22,36,21,35)(18,25,37,43,27,33,26,42)(19,29,30,34,32,46,31,47)(24,39,28,45,40,48,44,41)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29164);