#121952

order	512
nilpotency class	3
isoclinism	isoclinic class of #121952
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

389 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29148);