#94507

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄×C₃₂	C₂²
commutator	C₂	C₄×C₆₄
socle, prime radical	C₂²	C₄×C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

Polycyclic presentation:

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277990
maximal normal subgroup of	#72277990
maximal quotient of	#72277990, #72278050

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1703);