#121939

order	512
nilpotency class	3
isoclinism	isoclinic class of #121934
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	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

374 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,e^2=i,f^2=hi,g^2=hi,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=fhi,g^b=gh,f^c=fhi,g^c=gh

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, 29135);