#121025

order	512
nilpotency class	3
isoclinism	isoclinic class of #10374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28221);