#94504

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₈×C₃₂ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄×C₃₂	C₂²
commutator	C₂	C₈×C₃₂
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

78 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72277936, #72277944
maximal subgroup of	C₈×M₁₂₈, #72234395, #72277893, …
maximal normal subgroup of	C₈×M₁₂₈, #72234395, #72277893, …
maximal quotient of	#72234393, #72277918

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1700);