#121954

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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 384 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,e^2=h,f^2=hi,g^2=i,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.

Relationships

maximal subgroup of	#72286739, #72286796, #72286836
maximal normal subgroup of	#72286739, #72286796, #72286836

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29150);