#94216

order	512
nilpotency class	5
isoclinism	isoclinic class of #94206
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂³×C₁₆	C₂²
prime radical	C₂⁴	Q₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂³×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,19,16,22,39,23)(11,24,26,59,29,49,30,68)(12,31,34,57,36,20,55,54)(14,40,37,52,35,18,48,51)(15,42,38,21,41,50,33,56)(17,43,44,25,47,61,53,32)(27,58,62,46,64,70,65,71)(28,66,60,69,63,72,67,45);(1,3)(2,5)(4,8)(6,7)(9,11,14,27,41,63,55,43,39,26,48,62,38,67,36,25,16,29,35,64,15,28,34,61,13,30,37,65,33,60,12,32)(10,17,20,45,56,71,51,59,19,53,54,72,42,70,40,24,22,47,31,69,21,46,52,68,23,44,57,66,50,58,18,49)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgh,d^2=i,e^2=fg,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,h^a=hi,c^b=ci,d^b=dg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1412);