#121929

order	512
nilpotency class	3
isoclinism	isoclinic class of #121929
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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 372 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286788, #72286796, #72286824
maximal normal subgroup of	#72286788, #72286796, #72286824

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29125);