#121152

order	512
nilpotency class	3
isoclinism	isoclinic class of #10690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#401
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1443
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

483 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 483 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28348);