#94508

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₄×C₃₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₄²
commutator	C₄	C₄×C₃₂
Frattini, cosocle	C₄×C₃₂	C₂²
socle, prime radical	C₂²	#1397
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

66 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^2=i,f^2=g,g^2=h,h^2=i,b^a=be,d^a=di,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230689, #72230752, #72230755, …
maximal normal subgroup of	#72230689, #72230752, #72230755, …
maximal quotient of	#72230728, #72277990

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1704);