#93747

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	D₁₆
commutator	C₁₆	C₂×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂²
socle	C₂²	#1331
prime radical	C₂×D₁₆	C₈
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(17,18)(19,23)(20,32)(21,35)(22,27)(24,28)(25,29)(26,33)(30,42)(31,38)(34,48)(36,44)(37,43)(39,47)(40,45)(41,46);(18,23)(19,28)(20,29)(21,33)(24,25)(26,42)(27,38)(30,47)(31,45)(32,36)(34,46)(35,37)(39,48)(40,41)(43,44)

Polycyclic presentation:

2^9|a^2=d,c^2=eg,d^2=f,e^2=gi,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=ce,e^b=eg,g^b=gi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226994, #72227073, #72227078
maximal normal subgroup of	#72226994, #72227073, #72227078
maximal quotient of	#72226986

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 943);