#121924

order	512
nilpotency class	3
isoclinism	isoclinic class of #121705
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

409 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2,5,8,4,7,11,15)(3,6,10,14,9,13,12,16)(18,23)(19,27)(20,21)(24,32)(28,31)(29,30);(1,3,4,9)(2,6,7,13)(5,12,11,10)(8,16,15,14)(17,18,20,26,22,23,21,25)(19,28,27,24,29,32,30,31);(2,7)(5,11)(6,13)(10,12)(17,19,20,27,22,29,21,30)(18,24,26,32,23,31,25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=deh,e^2=i,f^2=i,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=fi,g^b=gh,f^c=fi,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, 29120);